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/*.aux
/*.fdb_latexmk
/*.fls
/*.log
/*.out
/*.pdf
/*.synctex.gz
/*.xdv
/*.bbl
/*.blg
mainNotes.bib
main.pdf

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#import "@preview/physica:0.9.5": pdv, super-T-as-transpose
#show: super-T-as-transpose
= Appendix
#include "predmode.typ"
The center principle is to assign the Raman tensor (i.e., change of polarizability caused by atomic displacement)
to each atom in the unit cell.
This including the following steps:
- Write out the change of polarizability caused by displacement of Si atom in A and C layer,
Where unknown non-zero components are denoted by $a_1$, $a_2$, $a_5$, $a_6$.
For example, when we move the Si atom in A layer slightly towards the x+ direction in $d$ distance,
the change of polarizability should be $mat(,a_2,a_1;a_2,,;a_1,,)d$.
This could be done by conclusion above.
- The Si atom in B layer have similar local environment as the A and C layer, with only a little difference.
We denote these difference by $epsilon_1$, $epsilon_2$, $epsilon_5$, $epsilon_6$,
and the absolute value of $epsilon_i$ should be much smaller than $a_i$.
For example, when we move the Si atom in B layer slightly towards the x+ direction in $d$ distance,
the change of polarizability should be $mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,)d$.
- The local environment of C atom in A layer is similar to the Si atom in A layer with charge reversed and
the system reversed along xy plane.
We denote these difference by $eta_1$, $eta_2$, $eta_5$, $eta_6$,
and the absolute value of $epsilon_i$ should be much smaller than $a_i$.
For example, when we move the C atom in A layer slightly towards the x+ direction in $d$ distance,
the change of polarizability should be $mat(,a_2+eta_2,-a_1-eta_1;a_2+eta_2,,;-a_1-eta_1,,)d$.
- Similar to the case in Si atoms, we derive the change of polarizability
caused by moving C atom in B layer slightly towards the x+ direction in $d$ distance,
which should be $mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,)d$.
Lets assign Raman tensor onto each atom.
That is, Raman tensor is derivative of the polarizability with respect to the atomic displacement:
$
alpha = pdv(chi, u)
$
where $u$ should be the displacement of the atom corresponding to a phonon mode.
But, even when $u$ is *NOT* the displacement of a phonon
(for example, lets only slightly move Si atom in A layer, keeping other atoms fixed),
the (high-frequency) polarizability is still well-defined,
and the will still cause a change in the polarizability.
Even more, the group representation theory is still applicable in this condition:
the only thing that matters is, when applying $g$ to the system,
the tensor transformed into $g^(-1) alpha g$ or $g alpha g^(-1)$,
no matter $alpha$ is Raman tensor or something else, or it is related to a phonon or not.
Thus, we can, in principle, "assign" Raman tensor of a phonon, to each atom.
This "assign" is unique since both the atom movement and all phonons have 24 dimensions.
Next, we consider what these single-atom-caused "Raman tensors" looks like.
For example, what happens if we move the Si atom in A layer slightly along the x+ direction?
Consider also move the Si atom in C layer slightly, along x+ or x- direction.
How about the Raman tensor caused by the both two atoms?
In first case, this is B2 representation in E1 representation. Thus the Raman tensor should be something like:
$
mat(,,2a_1;,,;2a_1,,;)
$
In the second case, it is A2 in E2. It turns out:
$
mat(,2a_2,;2a_2,,;,,;)
$
The average of these two tensors should be the s"Raman tensor" cause by move only the Si atom in A layer,
slightly towards x+ direction.
$
mat(,a_2,a_1;a_2,,;a_1,,;)
$
The difference should be the "Raman tensor" of the second atom.
$
mat(,-a_2,a_1;-a_2,,;a_1,,;)
$
// This approach applied relied on the fact that, all Si atom in 4H-SiC is "distinguishable" by the symmetry operations.
// I mean, what will happen if we have two Si atoms in A layer?
// Apparently, we could not extract the "Raman tensor" of only one of the two atoms.
// This is the case for the 6H-SiC.
// Hence, we will provide a more general approach to estimate the "Raman tensor" of a single atom.
Consider the Si atom in the B1 layer.
It lives in an environment quite similar to the A layer.
Thus, the "Raman tensor" caused by it should be similar to the one caused by the A layer:
$
mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)
$
Similar to the Si atom in B2 layer:
$
mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)
$
Same approach applied for Si atom vibrate in y direction.
When we move both Si atoms in A and C layer in y+ direction,
it is B1 in E1, thus the "Raman tensor" should be:
$
mat(,,;,,2a_3;,2a_3,;)
$
And if we move Si in A layer towards y+ but Si in C layer towards y-,
it is A2 in E2:
$
mat(2a_4,,;,-2a_4,;,,;)
$
Thus we get the "Raman tensor" of Si atom in A layer sololy move towards y+ direction:
$
mat(a_4,,;,-a_4,a_3;,a_3,;)
$
and the "Raman tensor" of Si atom in C layer towards y+ direction:
$
mat(-a_4,,;,a_4,a_3;,a_3,;)
$
Same applied for the Si atom in B layer:
$
mat(a_4+epsilon_4,,;,-a_4-epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)
$
$
mat(-a_4-epsilon_4,,;,a_4+epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)
$
Before consider z-direction, it is important to note that, $a_1$ $a_2$ $a_3$ $a_4$ are not independent.
Consider vibration along x+ direction (lets say the distance is $d$).
System energy caused by external electric field and vibration is:
$
E^T (mat(,,2a_1;,,;2a_1,,) d) E
$
Apply C#sub[3] to atom vibration and external field, energy should not change. We got:
$
(mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1)E)^T ( mat(,,2a_1;,,;2a_1,,)(-1/2 d) + mat(,,;,,2a_3;,2a_3,)(sqrt(3)/2 d) )
(mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1)E)
$
It is equal to:
$
E^T (mat(,,1/2 a_1 + 3/2 a_3;,,sqrt(3)/2 a_1 - sqrt(3)/2 a_3;1/2 a_1 + 3/2 a_3,sqrt(3)/2 a_1 - sqrt(3)/2 a_3,) d) E
$
Thus:
$
1/2 a_1 + 3/2 a_3 = 2a_1 #linebreak()
sqrt(3)/2 a_1 - sqrt(3)/2 a_3 = 0
$
Thus $a_1 = a_3$.
Apply the same method, we get $abs(a_2) = abs(a_4)$.
Since we have not define the sign of $a_4$, we could take $a_2 = a_4$.
Same for $epsilon$.
Now consider what if we move the Si atom in A layer along z+ direction.
If we move the Si atom in C layer along z+ direction, it is A1:
$
mat(2a_5,,;,2a_5,;,,2a_6;)
$
If we move the Si atom in C layer along z- direction, it is B1:
$
0
$
Thus we get the "Raman tensor" of Si atom in A or C layer towards z+ direction:
$
mat(a_5,,;,a_5,;,,a_6;)
$
Lets consider the C atom in A layer.
It should be somehow similar to the Si atom in A layer, but with a negative sign in some places,
and then add or subtract some little value.
Actually, the "transformation" of Si atom in A layer to C atom in A layer applied in the following steps:
- reverse charge.
- reverse system along xy plane.
First we consider the first step.
Taking the define of electricity tenser:
$
P = chi E
$
Lets reverse charge of the system, say we now have electricity tensor $chi'$. We get:
$
-P = chi'(-E)
$
Thus we get $chi' = chi$, the first step does not change the electricity tensor, nor the "Raman tensor".
Now we consider the second step.
For electricity tensor, it will become:
$
mat(1,,;,1,;,,-1) chi mat(1,,;,1,;,,-1)
$
For $u$, when it is along x or y direction, it will not change. When it is along z direction, it will become $-u$.
So in conclusion, Raman tensor of C atom in A layer could be estimated from the Raman tensor of Si atom in A layer, by:
- for movement alone x and y direction, xz yz should be applied a negative sign.
- for movement alone z direction, xx xy yy zz should be applied a negative sign.
Export "Raman tensor" of C atom in C layer from C atom in A layer, in the same way.
Now consider the C atom in B1 layer.
Is it similar to the C atom in A layer, just like that for Si atom?
No. It turns out to be similar to the C atom in C layer.
We summarize these stuff into @table-singleatom.
Until now, we only consider the "Raman tensor" caused by single atom or atoms move in the same amplitudes.
However, that is not the case in real phonon.
- In some A1 modes, only Si or C atom moves. If we take the magnitude of eigenvector as 1,
then amplitude of each atom is $1/(4sqrt(m_#text[Si]))$ or $1/(4sqrt(m_#text[C]))$.
- In other cases, the amplitude of Si and C are in the ration of $m_#text[C] : m_#text[Si]$.
thus the amplitude of Si atom is $1/2 sqrt(1/(m_#text[Si]+m_#text[Si]^2/m_#text[C]))$, so do the C atom.
Furthermore, we list predicted modes and their Raman tensors, in @table-predmode.
- $a$: Raman tensor of Si atom in A layer, large value.
- $epsilon$: Difference of Raman tensors of Si atom in A and B1 layer, small value.
- $eta$: Difference of Raman tensors of C and Si atom in A layer, small value.
- $zeta$: Difference of Raman tensors of C atoms in A and B layer, small value.
#page(flipped: true)[#figure({
table(columns: 4, align: center + horizon, inset: (x: 3pt, y: 5pt),
[*Move Direction*], [x], [y], [z],
[Si A], [$mat(,a_2,a_1;a_2,,;a_1,,;)$], [$mat(a_2,,;,-a_2,a_1;,a_1,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$],
[C A], [$mat(,a_2+eta_2,-a_1-eta_1;a_2+eta_2,,;-a_1-eta_1,,;)$],
[$mat(a_2+eta_2,,;,-a_2-eta_2,-a_1-eta_1;,-a_1-eta_1,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
[Si B1], [$mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)$],
[$mat(a_2+epsilon_2,,;,-a_2-epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
[$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
[C, B1], [$mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
[$mat(-a_2-eta_2-zeta_2,,;,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
[$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
[Si C], [$mat(,-a_2,a_1;-a_2,,;a_1,,;)$], [$mat(-a_2,,;,a_2,a_1;,a_1,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$],
[C, C], [$mat(,-a_2-eta_2,-a_1-eta_1;-a_2-eta_2,,;-a_1-eta_1,,;)$],
[$mat(-a_2-eta_2,,;,a_2+eta_2,-a_1-eta_1;,-a_1-eta_1,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
[Si B2], [$mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)$],
[$mat(-a_2-epsilon_2,,;,a_2+epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
[$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
[C, B2], [$mat(,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;a_2+eta_2+zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
[$mat(a_2+eta_2+zeta_2,,;,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
[$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
)},
caption: ["Raman tensor" caused by single atom],
placement: none,
)<table-singleatom>]

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// Raman Tensor for A1: line1 xx/yy; line2 zz
// Raman Tensor for E1: x-dirc xz or y-dirc yx
// Raman Tensor for E2: x-dirc xy or y-dirc xx or y-dirc -yy
// TODO: remove LO TO or not?
#page(flipped: true)[#figure({
let m(n, content) = table.cell(colspan: n, content);
let m2(content) = table.cell(colspan: 2, content);
let m3(content) = table.cell(colspan: 3, content);
let m4(content) = table.cell(colspan: 4, content);
set text(size: 9pt);
set par(justify: false);
table(columns: 11, align: center + horizon, inset: (x: 3pt, y: 5pt),
[*Representation in C#sub[6v]*], m3[A#sub[1]], m3[E#sub[1]], m4[E#sub[2]],
[*Relative Vibration Direction*],
[Si: $+-+-$ #linebreak() C: $0000$], [Si: $0000$ #linebreak() C: $+-+-$], [Si: $++++$ #linebreak() C: $----$],
[Si: $+-+-$ #linebreak() C: $-+-+$], [Si: $+-+-$ #linebreak() C: $+-+-$], [Si: $++++$ #linebreak() C: $----$],
[Si: $++--$ #linebreak() C: $-++-$], [Si: $+--+$ #linebreak() C: $++--$],
[Si: $++--$ #linebreak() C: $+--+$], [Si: $+--+$ #linebreak() C: $--++$],
[*Vibration Direction*], m3[z], m3[x/y], m4[x/y],
[*Raman Tensor Predicted*], [xx/yy: $-2A_#text[Si] epsilon_5$ #linebreak() zz: $-2A_#text[Si]epsilon_6$],
[xx/yy: $-2A_#text[C]zeta_5$ #linebreak() zz: $-2A_#text[C]zeta_6$],
[xx/yy: $2A_#text[Si] (2a_5+epsilon_5) + 2A_#text[C] (2a_5+eta_5+zeta_5)$ #linebreak() zz: $2A_#text[Si] (2a_6+epsilon_6) + 2A_#text[C] (2a_6+eta_6+zeta_6)$],
[xz/yz: $-2A_#text[Si]epsilon_1-2A_#text[C]zeta_1$],
[xz/yz: $-2A_#text[Si]epsilon_1+2A_#text[C]zeta_1$],
[xz/yz: $2A_#text[Si] (2a_1+epsilon_1) +2A_#text[C] (2a_1+2eta_1+zeta_1))$],
[xx/-yy/xy: $2A_#text[Si] (2a_2+epsilon_2) -2A_#text[C] (2a_2+2eta_2+zeta_2))$],
[xx/-yy/xy: $-2A_#text[Si]epsilon_2-2A_#text[C]zeta_2$],
[xx/-yy/xy: $2A_#text[Si] (2a_2+epsilon_2) +2A_#text[C] (2a_2+2eta_2+zeta_2))$],
[xx/-yy/xy: $-2A_#text[Si]epsilon_2+2A_#text[C]zeta_2$],
[*Raman Intensity Predicted*], m2[weak], [strong], m2[weak], [strong], m2[weak], [strong], [weak],
[*Raman Tensor Calculated*],
[-1.68 #linebreak() 1.34], [0.10 #linebreak() -1.33], [-7.68 #linebreak() 21.65],
[-1.56], [-0.30], [7.32], [-0.41], [1.06], [9.41], [-0.71],
// [*x*], [1 axial acoustic], [0 axial optical], [1 axial optical],
// [0 axial acoustic], [1 axial optical], [1 axial optical],
// m2[0.5 acoustic], m2[0.5 optical],
[*Type*], [axial acoustic], [axial optical], [longitudinal optical],
[planer acoustic], [planer optical], [transverse optical],
m2[planer acoustic], m2[planer optical],
[*Move-towards Atom-pairs* (In-plane/Out-plane)], [4/0], [0/4], [4/4], [0/4], [4/0], [4/4], [0/2], [2/0], m2[4/2],
// [*Predicted Frequency*], [low], [medium], [high], [medium], [low], [high], [low], [medium], m2[high],
[*Calculated Frequency*],
[591.90], [812.87], [933.80], [257.35], [746.91], [776.57], [190.51], [197.84], [756.25], [764.33]
)},
caption: [Predicted modes and their "Raman tensor"],
placement: none,
)<table-predmode>]

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= Introduction
4H-SiC 性能很好、器件应用广泛，因此需要开发原位的、非破坏性的表征技术。
4H-SiC 是一种具有优良性质的半导体材料，包括宽禁带、高临界电场强度、高热导率和沿 c 轴的高电子迁移率，因此受到了广泛的研究。
受益于外延技术的发展和新能源产业的需求增长，4H-SiC 在功率电子器件中得到了广泛应用。
然而，4H-SiC 器件的性能仍然受到缺陷的限制，如何在生长和工作条件下避免缺陷的产生仍然是一个挑战。
此外，用于表征 SiC 掺杂的方案（如二次离子质谱（SIMS）和霍尔效应测量）通常是破坏性的且耗时，因此迫切需要开发原位和非破坏性的表征技术。
// 随着尺寸缩小，4H-SiC 中的缺陷对器件性能的影响更加显著。（纳米材料？
The 4H-silicon carbide (SiC) has long attracted a lot of research
thanks to its wider bandgap, higher critical electric field strength,
higher thermal conductivity, and higher electron mobility along the c-axis
than silicon (Si) and gallium arsenide (GaAs) as well as other SiC polytypes
@casady_status_1996 @okumura_present_2006.
It has been widely used in power electronic devices
thanks to the development of epitaxy technology and the increasing application in the new energy industry
@tsuchida_recent_2018 @harada_suppression_2022 @sun_selection_2022.
However, the performance of 4H-SiC devices remains constrained by the presence of defects,
which may be introduced during the growth process @nishio_triangular_2020 @demenet_tem_2005
or arise under device operating conditions
@mahadik_ultraviolet_2012 @okada_dependences_2018 @iwahashi_extension_2017 @caldwell_driving_2010
@miyanagi_annealing_2006 @iijima_correlation_2017.
Furthermore,
conventional methods for characterizing the doping properties of SiC
(such as secondary ion mass spectrometry (SIMS) @kudriavtsev_quantitative_2003 @kim_characteristics_2024
and Hall effect measurements @noguchi_comparative_2021 @asada_hall_2016),
are destructive and time-consuming.
Therefore, there is a pressing need to develop an in-situ and non-destructive characterization techniques.
拉曼主要体现声子的信息，并且早已经有应用，主要用来区分 SiC 的多型。
声子（量子化的原子振动）在理解晶体的原子结构以及热电性质方面起着重要作用。
声子可以通过多种实验技术来探测，包括 EELS、IR 吸收谱等。
拉曼光谱是最常用的方法，它提供了一种无损、非接触、快速和局部的声子测量方法。
早有关于拉曼的研究，且拉曼已被广泛用于区分 SiC 的多型。
Phonons (quantized atomic vibrations) play a fundamental role
in understanding the atomic structure as well as the thermal and electrical properties of semiconductors.
They could be probed by various experimental techniques,
such as electron energy loss spectroscopy @yan_single-defect_2021 @egoavil_atomic_2014
and infrared absorption spectroscopy @pluchery_infrared_2012 @tong_temperature-dependent_2018.
Among these techniques,
Raman spectroscopy is the most commonly used method,
// TODO: 增加一些引用，可以先不用收集文献给这里，最后把其它部分的全拿过来就行了。
as it provides non-destructive, non-contact, rapid and spatially localized measurement of phonons
that near the #sym.Gamma point in reciprocal space.
Studies in Raman scattering of 4H-SiC have been conducted since as early as 1968 @feldman_phonon_1968
and nowadays have been widely employed to identification of different SiC polytypes
@guo_characterization_2012 @yan_study_2016 @hundhausen_characterization_2008 @nakashima_raman_2013.
拉曼谱中有更多信息。有一些新的研究，但他们还有不足。
近年来，更多信息被从拉曼光谱中挖掘出来。
LO 声子峰或 LOPC 峰已经被证明与自由载流子的类型和浓度有关，它们已经被用于估计离子注入层的厚度和 n 型 SiC 的掺杂浓度。
3C-SiC 的一类层错的拉曼和 EELS 光谱已经被研究，但更多类型的层错及其在 4H-SiC 中的拉曼光谱还没有被研究。
有人提出了可能的 N 掺杂原子的拉曼峰，但并没有在实验上对比验证，同时 Al 或其它点缺陷的拉曼峰也缺少系统的研究。
此外，拉曼光谱上仍有一些不知来源的峰；同时，也缺少一些理论上预测应该存在的峰。
// TODO: 总结更多文献
In recent years, increasingly rich information has been extracted from Raman spectra of 4H-SiC.
Longitudinal optical phonon–plasmon coupling (LOPC) peek
has been utilized to rapidly estimate the doping concentration in n-type SiC @harima_raman_1995
and to identify doping type in different layers @song_depth_2020,
while the influence of free carriers on the LO phonon peek in p-type SiC
has not been systematically investigated yet.
Peeks associated with one type of stacking faults in 3C-SiC have been investigated @yan_single-defect_2021,
while other types of stacking faults and their Raman spectra in 4H-SiC remain unexplored.
Peeks associated with nitrogen dopants have been proposed @_n-sic_2010
but have not been experimentally verified,
while systematic studies on the Raman spectra of aluminum or other point defects are still lacking.
Moreover, certain phonon modes predicted by theory remain unobserved,
while there are still some unidentified peaks in the Raman spectra.
本文通过三种方式，研究 4H-SiC 中带缺陷和不带缺陷的声子。我们第一次做到了什么什么。
In this paper, we explored the phonon in 4H-SiC by three ways:
symmetry analysis, first-principles calculations, and Raman experiment.
We first investigated the phonon modes in perfect 4H-SiC,
and then explored the phonon modes associated with defects and doping.
// TODO: 描述自己做了什么，强调自己是第一次做到了什么。

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\documentclass[preprintnumbers,aps,prl,nofootinbib]{revtex4-2}
% \linespread{1.5}
\usepackage{graphicx}
\usepackage{dcolumn}
\usepackage{bm}
\usepackage[mathlines]{lineno}
% \usepackage[slantfont, boldfont]{xeCJK}
% \setCJKmainfont{Microsoft YaHei}
% \setCJKmonofont{Source Code Pro}
% \setCJKsansfont{YouYuan}
\linenumbers
\renewcommand\linenumberfont{\normalfont}
\usepackage{array}
\usepackage{svg}
\usepackage{rotating}
\usepackage{amsmath}
\usepackage{tabularray}
\usepackage{siunitx}
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\usepackage{soul}
\usepackage[svgnames]{xcolor}
% \newcommand{\del}[1]{\textcolor{red}{\st{#1}}}
% \DeclareRobustCommand{\add}[1]{{\sethlcolor{LightGreen}\hl{#1}}}
% \usepackage[colorlinks]{hyperref}
% \usepackage[style=cms]{citation-style-language}
% \addbibresource{ref.bib}
\begin{document}
\preprint{APS/123-QED}
\title{Title Title Title}
\author{Haonan Chen}\altaffiliation{Physics Department, XYZ University.}
\begin{abstract}
An article usually includes an abstract, a concise summary of the work
covered at length in the main body of the article.
\end{abstract}
\maketitle
\section{Introduction}\label{sec_introduction}
% SiC 是很好的材料。
% 其中，4H-SiC 是SiC的一种多型，它的性质更好，近年来随着外延工艺的成熟而获得了更多的关注。
% SiC 中的声子与材料的性质密切相关。通常来说，通过拉曼来区分多型。我们相信可以通过声子来挖掘更多的信息。
SiC is a promising wide-bandgap semiconductor material
with high critical electric field strength and high thermal conductivity.
It has been widely used in power electronic devices and has long attracted a lot of research
\cite{casady_status_1996, okumura_present_2006}.
% The 4H-SiC has a wider bandgap, higher critical electric field strength,
% higher thermal conductivity, and higher electron mobility along the c-axis than other polytypes.
% Currently, the 4H-SiC has gradually received more attention than other polytypes,
% thanks to the development of epitaxy technology and the increasing application in the new energy industry
% \cite{tsuchida_recent_2018, harada_suppression_2022, sun_selection_2022}.% TODO: 多引用一些近年来的文献，有很多
%
% Currently, the 4H-SiC has gradually received more attention than other polytypes,
% thanks to the development of epitaxy technology and the increasing application in the new energy industry
% \cite{tsuchida_recent_2018, harada_suppression_2022, sun_selection_2022}.
% 某某人做了什么
% 4H-SiC is a promising wide-bandgap semiconductor material
% with high critical electric field strength and high thermal conductivity.
% It has been widely used in power electronic devices and has long attracted a lot of research
% \cite{casady_status_1996, okumura_present_2006}.
% SiC has more than 250 polytypes \cite{cheung_silicon_2006}, the 3C-SiC has been widely studied in the past decades
% 139 \cite{dompoint_kinetics_2011, izhevskyi_review_2000, kimoto_bulk_2016, tang_atomic_2007, blumenau_effect_2005,
% 140 bernardini_interaction_2005, rodney_ab_2017, blumenau_straight_2002, blumenau_structure_2003,
% 141 vashishta_interaction_2007}.
%
% 147 It shows an exclusive application in power electronic devices,
% 148 such as the Schottky barrier diodes and the power metal-oxide-semiconductor field-effect transistors
% 149 \cite{bhatnagar_comparison_1993, kimoto_material_2015}.
% 150 However, these applications are negatively affected by Shockley stacking faults (SSFs),
% 151 which are a class of common defects in 4H-SiC.
% 152
% 153 The structure of SiC could be considered as a stack of Si and C atomic layers,
% 154 and only the stacking sequence of exactly repeating A-B-C-B leads to the formation of 4H-SiC without defects.
% 155 The differences in formation energies between different stacking positions are only about
% 156 \qty{1}{\meV/atom} \cite{kimoto_bulk_2016},
SiC has many excellent properties and wide applications.
Phonons in SiC are important. They can influence the properties of SiC and can be used to characterize the materials.
There are many existing studies on phonons in SiC, but they have some shortcomings.
In this paper, we do some things. We do something for the first time.
\section{Methods}\label{sec_methods}
\section{Results}\label{sec_results}
\subsection{Phonons in Perfect 4H-SiC}
% 拉曼活性的声子模式对应于 Gamma 点附近的声子模式。
% 根据这些声子模式在拉曼实验中的表现，我们将这些声子分成三个部分。
Raman scattering peeks correspond to phonons located near $\Gamma$ point in reciprocal space.
We classified these phonons into three categories according to their behavior in Raman scattering:
(1) phonons could not be observed in Raman scattering spectrum,
either because they are Raman inactive or their scattering intensity is too weak;
(2) phonons could be observed in Raman scattering spectrum and with weak or no polarities,
their frequencies were independent of the direction of the incident light;
(3) strong polar phonons,
which were visible in Raman scattering spectrum,
and their frequencies depend on the direction of the incident light.
% 我们计算了 4H-SiC 在 A-Gamma 和 Gamma-M 上的声子频率，如图和附录1所示。
% 在拉曼散射中，起作用的模式都是那些非常接近于 Gamma 的模式
% （如图中的点所示，分为位于 1/50 和 1/100 处，这两条线分别对应于拉曼散射在 z 方向入射/散射和 y 方向入射/散射）。
% 大多数声子模式在 Gamma 附近都是连续的，这使得它们的频率对入射光的方向不敏感；
% 然而，少数声子具有较强的极性，这使得声子之间存在长程的库伦相互作用（引用文献），并导致 gamma 附近的频率不同，如图中的某两条线所示。
% 据此，我们将无缺陷的 4H-SiC 的声子分成三类：
% 无拉曼活性或拉曼散射强度太弱的模式，它们在拉曼散射谱上不可见；
% 拉曼散射强度足够大且极性不强的模式，它们在拉曼散射谱上可以看到，且频率与拉曼入射光方向无关；
% 极性声子，它们在拉曼散射谱上可以看到，不仅频率与入射光方向有关，而且可与载流子发生一些相互作用。
Phonons in defect-free 4H-SiC are calculated at A-$\Gamma$ and $\Gamma$-M,
as shown in Figure \ref{fig:phonon} and Table \ref{tab:phonon}.
Raman active phonons are very close to $\Gamma$,
as indicated by the points in the figure.
Because of the consistency of the most phonon modes near $\Gamma$,
most of the phonon frequencies are insensitive to the direction of the incident light.
However, some phonons have strong polarities,
which leads to long-range Coulomb interactions between phonons,
and results in different frequencies near $\Gamma$,
as shown by the two lines in the figure.
Thus, we divide the phonons of defect-free 4H-SiC into three categories:
(1) Raman inactive or too weak Raman intensity,
which are invisible in the Raman scattering spectrum;
(2) Raman active phonons with strong polarities,
which are visible in the Raman scattering spectrum,
and their frequencies are independent of the direction of the incident light;
(3) Polar phonons,
which are visible in the Raman scattering spectrum,
and their frequencies depend on the direction of the incident light,
and can interact with carriers.
% insert fig1.svg
\begin{figure}[h]
\centering
\includegraphics{../画图/声子不连续/整体图.pdf}
\caption{Phonon dispersion of defect-free 4H-SiC.}
\label{fig:phonon}
\end{figure}
\subsubsection{}
\appendix
\section{A little more on appendixes}
\begin{sidewaystable}
\centering
{
\caption{Weak- and None-polarized phonons near $\Gamma$ point}
\begin{tblr}{
hlines,vlines,colsep=2pt,width=\textwidth,
colspec={
X[10,c,m]
*{3}{X[c,m]} % E2
*{3}{X[c,m]} % E2
*{2}{X[c,m]} % E1
*{2}{X[4,c,m]} % 2B1
*{2}{X[c,m]} X[2,c,m] % A1
*{2}{X[c,m]} % E1
*{3}{X[c,m]} % E2
*{3}{X[c,m]} % E2
*{2}{X[c,m]} X[2,c,m] % A1
*{2}{X[4,c,m]} % 2B1
}
}
\textbf{Direction of Incident \& Scattered Light}
& \SetCell[c=26]{}{Any direction \\ (not depend on direction of incident \& scattered light)}
& & & & & & & & & & & & & & & & & & & & & & & & &
\\
\textbf{Number of Phonon}
& 1 & \SetCell[c=2]{} 2 & % E2
& 3 & \SetCell[c=2]{} 4 & % E2
& 5 & 6 % E1
& 7 & 8 % 2B1
& \SetCell[c=3]{} 9 & & % A1
& 10 & 11 % E1
& 12 & \SetCell[c=2]{} 13 & % E2
& 14 & \SetCell[c=2]{} 15 & % E2
& \SetCell[c=3]{} 16 & & % A1
& 17 & 18 % 2B1
\\
\textbf{Vibration Direction}
& x & \SetCell[c=2]{} y & % E2
& x & \SetCell[c=2]{} y & % E2
& x & y % E1
& z & z % 2B1
& \SetCell[c=3]{} z & & % A1
& x & y % E1
& x & \SetCell[c=2]{} y & % E2
& x & \SetCell[c=2]{} y & % E2
& \SetCell[c=3]{} z & & % A1
& z & z % 2B1
\\
\textbf{Representation in Group $\mathrm{C_{6v}}$}
& \SetCell[c=3]{} $\mathrm{E_2}$ & & % E2
& \SetCell[c=3]{} $\mathrm{E_2}$ & & % E2
& \SetCell[c=2]{} $\mathrm{E_1}$ & % E1
& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
& \SetCell[c=3]{} $\mathrm{A_1}$ & & % A1
& \SetCell[c=2]{} $\mathrm{E_1}$ & % E1
& \SetCell[c=3]{} $\mathrm{E_2}$ & & % E2
& \SetCell[c=3]{} $\mathrm{E_2}$ & & % E2
& \SetCell[c=3]{} $\mathrm{A_1}$ & & % A1
& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
\\
\textbf{Representation in Group $\mathrm{C_{2v}}$}
& $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
& $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
& $\mathrm{B_2}$ & $\mathrm{B_1}$ % E1
& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
& \SetCell[c=3]{} $\mathrm{A_1}$ & & % A1
& $\mathrm{B_2}$ & $\mathrm{B_1}$ % E1
& $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
& $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
& \SetCell[c=3]{} $\mathrm{A_1}$ & & % A1
& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
\\
\textbf{Scattering in Polarization}
& xy & xx & yy % E2
& xy & xx & yy % E2
& xz & yz % E1
& - & - % 2B1
& xx & yy & zz % A1
& xz & yz % E1
& xy & xx & yy % E2
& xy & xx & yy % E2
& xx & yy & zz % A1
& - & - % 2B1
\\
\textbf{Raman Intensity (a.u.)}
& \SetCell[c=3]{} $0.17$ & & % E2
& \SetCell[c=3]{} $1.13$ & & % E2
& \SetCell[c=2]{} $2.43$ & % E1
& $0$ & $0$ % 2B1
& \SetCell[c=2]{} $2.83$ & & $1.79$ % A1
& \SetCell[c=2]{} $0.09$ & % E1
& \SetCell[c=3]{} $88.54$ & & % E2
& \SetCell[c=3]{} $0.50$ & & % E2
& \SetCell[c=2]{} $0.01$ & & $1.78$ % A1
& $0$ & $0$ % 2B1
\\
\textbf{Visible in Common Raman Experiment}
& \SetCell[c=3]{} Yes & & % E2
& \SetCell[c=3]{} Yes & & % E2
& \SetCell[c=2]{} Yes & % E1
& No & No % 2B1
& \SetCell[c=3]{} Yes & & % A1
& \SetCell[c=2]{} No & % E1
& \SetCell[c=3]{} Yes & & % E2
& \SetCell[c=3]{} No & & % E2
& \SetCell[c=2]{} No & & Yes % A1
& No & No % 2B1
\\
\textbf{Wavenumber (Simulation) ($\mathrm{cm^{-1}}$)}
& \SetCell[c=3]{} $190.51$ & & % E2
& \SetCell[c=3]{} $197.84$ & & % E2
& \SetCell[c=2]{} $257.35$ & % E1
& $389.96$ & $397.49$ % 2B1
& \SetCell[c=3]{} $591.90$ & & % A1
& \SetCell[c=2]{} $746.91$ & % E1
& \SetCell[c=3]{} $756.25$ & & % E2
& \SetCell[c=3]{} $764.33$ & & % E2
& \SetCell[c=3]{} $812.87$ & & % A1
& $885.68$ & $894.13$ % 2B1
\\
\textbf{Wavenumber (Experiment) ($\mathrm{cm^{-1}}$)}
& \SetCell[c=3]{} $195.5$ & & % E2
& \SetCell[c=3]{} $203.3$ & & % E2
& \SetCell[c=2]{} $269.7$ & % E1
& - & - % 2B1
& \SetCell[c=3]{} $609.5$ & & % A1
& \SetCell[c=2]{} - & % E1
& \SetCell[c=3]{} $776$ & & % E2
& \SetCell[c=3]{} - & & % E2
& \SetCell[c=2]{} - & $839$ % A1
& - & - % 2B1
\\
\textbf{Electrical Polarity}
& \SetCell[c=3]{} None & & % E2
& \SetCell[c=3]{} None & & % E2
& \SetCell[c=2]{} Weak & % E1
& None & None % 2B1
& \SetCell[c=3]{} Weak & & % A1
& \SetCell[c=2]{} Weak & % E1
& \SetCell[c=3]{} None & & % E2
& \SetCell[c=3]{} None & & % E2
& \SetCell[c=3]{} Weak & & % A1
& None & None % 2B1
\\
\end{tblr}
}
{
\caption{Strong-polarized phonons near $\Gamma$ point}
\begin{tblr}{
hlines,vlines,colsep=2pt,width=\textwidth,hspan=even,
colspec={
X[5,c,m]
X[c,m] % z x
X[c,m] % z y
*{2}{X[c,m]}X[c,m,1.5] % z z
X[c,m,2]X[c,m]X[c,m,2] % y z
X[c,m,1.5] % y x
X[c,m,2] % y y
*{3}{X[c,m]}X[c,m,1.5] % 45 y&z mainly z
X[c,m,1.5] % 45 x
*{4}{X[c,m]} % 45 y&z mainly y
}
}
\textbf{Direction of Incident \& Scattered Light}
& \SetCell[c=5]{} z & & & &
& \SetCell[c=5]{} y & & & &
& \SetCell[c=9]{} between z and y, 10\textdegree{} to z & & & & & & & &
\\
\textbf{Number of Phonon}
& 1 & 2 % z E1
& \SetCell[c=3]{} 3 & & % z A1
& \SetCell[c=3]{} 1 & & % y z
& 2 % y x
& 3 % y y
& \SetCell[c=4]{} 1 & & & % 45 y&z mainly z
& 2 % 45 x
& \SetCell[c=4]{} 3 & & & % 45 y&z mainly y
\\
\textbf{Vibration Direction}
& {x \\ (TO)} & {y \\ (TO)} % z E1
& \SetCell[c=3]{} z (LO) & & % z A1
& \SetCell[c=3]{} z (TO) & & % y z
& {x \\ (TO)} % y x
& {y \\ (LO)} % y y
& \SetCell[c=4]{} {y-z mixed \\ (LO-TO mixed)} & & & % 45 y&z mainly z
& x (TO) % 45 x
& \SetCell[c=4]{} {y-z mixed \\ (LO-TO mixed)} & & & % 45 y&z mainly y
\\
\textbf{Representation in Group $\mathrm{C_{6v}}$}
& \SetCell[c=2]{} $\mathrm{E_1}$ &
& \SetCell[c=3]{} $\mathrm{A_1}$ & &
& \SetCell[c=14]{} Not applicable & & & & & & & & & & & & &
\\
\textbf{Representation in Group $\mathrm{C_{2v}}$}
& $\mathrm{B_2}$ & $\mathrm{B_1}$ % z E1
& \SetCell[c=3]{} $\mathrm{A_1}$ & & % z A1
& \SetCell[c=3]{} $\mathrm{A_1}$ & & % y z
& $\mathrm{B_2}$ % y x
& $\mathrm{B_1}$ % y y
& \SetCell[c=4]{} Not Applicable & & & % 45 y&z mainly z
& $\mathrm{B_2}$ % 45 x
& \SetCell[c=4]{} Not Applicable & & & % 45 y&z mainly y
\\
\textbf{Scattering in Polarization}
& xz & yz % z E1
& xx & yy & zz % z A1
& xx & yy & zz % y z
& xz % y x
& yz % y y
& xx & yy & yz & zz % 45 y&z mainly z
& xz % 45 x
& xx & yy & yz & zz % 45 y&z mainly y
\\
\textbf{Raman Intensity (a.u.)}
& \SetCell[c=2]{} $53.52$ & % z E1
& \SetCell[c=2]{} $58.26$ & & $464.69$ % z A1
& \SetCell[c=2]{} $56.86$ & & $454.09$ % y z
& $53.52$ % y x
& $53.55$ % y y
& \SetCell[c=2]{} $53.71$ & & $3.20$ & $425.98$ % 45 y&z mainly z
& $53.56$ % 45 x
& \SetCell[c=2]{} $3.60$ & & $50.36$ & $27.99$ % 45 y&z mainly y
\\
\textbf{Visible in Common Raman Experiment}
& \SetCell[c=2]{} Yes & % z E1
& \SetCell[c=2]{} {Yes \\ (LOPC)} & & No % z A1
& Yes (overfocused) & No & Yes (overfocused) % y z
& Yes % y x
& {Yes \\ (LOPC)} % y y
& \SetCell[c=4]{} ??? & & & % 45 y&z mainly z
& ??? % 45 x
& \SetCell[c=4]{} ??? & & & % 45 y&z mainly y
\\
\textbf{Wavenumber (Simulation) ($\mathrm{cm^{-1}}$)}
& \SetCell[c=2]{} $776.57$ & % z E1
& \SetCell[c=3]{} $933.80$ & & % z A1
& \SetCell[c=3]{} $761.80$ & & % y z
& $776.57$ % y x
& $941.33$ % y y
& \SetCell[c=4]{} $762.76$ & & & % 45 y&z mainly z
& $776.57$ % 45 x
& \SetCell[c=4]{} $940.86$ & & & % 45 y&z mainly y
\\
\textbf{Electrical Polarity} & \SetCell[c=19]{} Strong & & & & & & & & & & & & & & & & & &
\end{tblr}
}
\label{tab:phonon}
\end{sidewaystable}
\bibliography{ref}
\end{document}

55
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@@ -0,0 +1,55 @@
#import "@preview/starter-journal-article:0.4.0": article, author-meta
#show: article.with(
title: "Article Title",
authors: (
"Haonan Chen": author-meta(
"xmu",
// email: "chn@chn.moe",
),
"Junyong Kang": author-meta(
"xmu",
email: "jykang@xmu.edu.cn"
)
),
affiliations: (
"xmu": "Xiamen University",
),
abstract: [#lorem(100)],
keywords: ("Typst", "Template", "Journal Article"),
)
// 行号
#set par.line(numbering: "1")
// 两端对齐
#set par(justify: true)
#set par(first-line-indent: (amount: 2em, all: true))
// 中文使用思源宋体，英文使用 Times New Roman
#set text(font: ("Times New Roman", "Source Han Serif SC"))
// 图表标题
#show figure.caption: it => {
set text(10pt)
align(center, box(align(left, it), width: 80%))
}
// 页码
#set page(
numbering: "1/1",
)
// TODO: why globally set placement not work?
// #set figure(placement: none)
// 标题序号
#set heading(numbering: "1.")
#include "introduction.typ"
#include "method.typ"
#include "result/default.typ"
#include "appendix/default.typ"
#include "others/default.typ"
#bibliography("./ref.bib", title: "Reference", style: "american-physics-society")

2
paper/mainNotes.bib
View File

@@ -1,2 +0,0 @@
@CONTROL{REVTEX42Control}
@CONTROL{apsrev42Control,author="08",editor="1",pages="0",title="0",year="1"}

21
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@@ -0,0 +1,21 @@
= Method
== Experiment details
外延片的厚度、掺杂浓度、生长 C/Si 比。
拉曼设备的型号。激光的波长，背散射。共焦针孔。对焦时向上或向下调整的距离。
画个图，表示正入射、掠入射和肩入射的角度关系。
Only back-scattering configurations were considered in this study.
== Simulation details
无缺陷的模型大小。带缺陷的情况下，模型大小，每个模型的代号和缺陷结构。
第一性原理计算使用 VASP，使用 PBE PAW，平面波截断能，K 点网格，涂抹，自洽和弛豫的 threshold。
声子计算使用 phonopy phono3py ufo，BEC 修正的算法。
// 在略入射的过程中，角度的关系不大（即使是 60 度入射，和 90 度入射差别也不大）。因此我们按照折射角 20 度的结果计算。
// 考虑到斜切，则大约是 24 度。

4
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@@ -0,0 +1,4 @@
= others
#include "掺杂晶格变化.typ"
#include "晶格变化导致的频率变化.typ"

16
paper/others/掺杂晶格变化.typ Normal file
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@@ -0,0 +1,16 @@
#figure({
set par(justify: false);
table(columns: 5, align: center + horizon,
table.cell(colspan: 2, rowspan: 2)[*Defect Type*],
table.cell(colspan: 3)[*Variation of Lattice Constant (#sym.permille)*], [x], [y], [z],
table.cell(rowspan: 2)[N#sub[C]], [h], [-0.47], [-0.48], [-0.70], [k], [-0.60], [-0.62], [-0.38],
table.cell(rowspan: 2)[Al#sub[Si]], [h], [0.71], [0.78], [1.02], [k], [0.72], [0.80], [1.01],
table.cell(rowspan: 2)[V#sub[C]], [h], [-0.88], [-1.85], [1.92], [k], [-0.47], [-1.02], [0.45],
table.cell(rowspan: 2)[V#sub[Si]], [h], [0.53], [0.64], [0.37], [k], [0.49], [0.58], [0.66],
table.cell(rowspan: 4)[N#sub[C]+C#sub[Si]],
[h, C#sub[3v]], [0.33], [1.44], [3.25], [k, C#sub[3v]], [0.28], [1.22], [3.51],
[h, C#sub[v]], [1.86], [0.60], [2.24], [k, C#sub[v]], [2.05], [1.10], [1.63],
)},
caption: [Variation of Lattice Constant caused by Defects],
placement: none,
)<table-dope>

28
paper/others/晶格变化导致的频率变化.typ Normal file
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@@ -0,0 +1,28 @@
#page(flipped: true)[#figure({
set par(justify: false);
table(columns: 12, align: center + horizon, // inset: (x: 3pt, y: 5pt),
table.cell(rowspan: 3)[*Variation of Lattice Constant*],
table.cell(colspan: 11)[*Variation of Frequency (cm#super[-1])*],
table.cell(colspan: 6)[*Negligible-polar Phonons*],
table.cell(colspan: 2)[*Strong-polar Phonons (z)*], table.cell(colspan: 3)[*Strong-polar Phonons (y)*],
[E#sub[2] at 195.5 cm#super[-1]], [E#sub[2] at 203.3 cm#super[-1]], [E#sub[1] at 269.7 cm#super[-1]],
[A#sub[1] at 609.5 cm#super[-1]], [E#sub[2] at 776 cm#super[-1]], [A#sub[1] at 839 cm#super[-1]],
[E#sub[1] at 795 cm#super[-1]], [A#sub[2] at 963 cm#super[-1]],
[z at 782 cm#super[-1]], [x at 796 cm#super[-1]], [y at ? cm#super[-1]],
[In-plane, expand $1 permille$], [0.00], [0.02], [0.05], [-0.75], [-2.21], [-1.17], [-2.14], [-1.30], [-1.00], [-2.14], [-2.35],
[In-plane, shrink $1 permille$], [0.00], [-0.01], [-0.04], [0.76], [2.21], [1.16], [2.13], [1.31], [1.02], [2.13], [2.35],
[Out-of-plane, expand $1 permille$], [-0.03], [-0.09], [0.02], [-0.95], [-0.58], [-1.64], [-0.49], [-1.69], [-1.55], [-0.49], [-0.69],
[Out-of-plane, shrink $1 permille$], [0.00], [0.08], [-0.05], [0.93], [0.57], [1.61], [0.49], [1.69], [1.55], [0.49], [0.69],
[x, expand $1 permille$], [-0.09], [-0.08], [-0.08], [-0.38], [-1.38], [-0.59], [-1.37], [-0.65], [-0.49], [-1.33], [-0.98],
[x, shrink $1 permille$], [-0.09], [-0.10], [-0.12], [0.38], [0.83], [0.58], [0.77], [0.65], [0.53], [1.32], [0.98],
)},
caption: ["Raman tensor" caused by single atom],
placement: none,
)<table-dope>]
// E2 E2 E1 2B1 A1
// m(3)[195.5], m(3)[203.3], m(2)[269.7], m(2)[-], m(3)[609.5],
// E1 E2 E2 A1 2B1
// m(2)[-], m(3)[776], m(5)[-], [839], m(2)[-],
// // z y 45 y&z
// m(2)[776.57], m(3)[933.80], m(3)[761.80], [776.57], [941.33], m(4)[762.76], [776.57], m(4)[940.86],

464
paper/ref.bib
View File

@@ -1,9 +1,147 @@
@article{demenet_tem_2005,
title = {{TEM} observations of the coexistence of perfect and dissociated dislocations in {SiC} under high stress},
volume = {2},
issn = {1610-1634},
url = {https://onlinelibrary.wiley.com/doi/10.1002/pssc.200460541},
doi = {10.1002/pssc.200460541},
language = {en},
number = {6},
urldate = {2022-07-04},
journal = {Physica Status Solidi C: Current Topics in Solid State Physics},
author = {Demenet, J. L. and Milhet, X. and Rabier, J.},
month = apr,
year = {2005},
pages = {1987--1991},
file = {Demenet 等。 - 2005 - TEM observations of the coexistence of perfect and.pdf:/home/chn/Zotero/storage/NHRXJKCK/Demenet 等。 - 2005 - TEM observations of the coexistence of perfect and.pdf:application/pdf},
}
@article{caldwell_driving_2010,
title = {On the driving force for recombination-induced stacking fault motion in {4H}-{SiC}},
volume = {108},
issn = {0021-8979, 1089-7550},
url = {http://aip.scitation.org/doi/10.1063/1.3467793},
doi = {10.1063/1.3467793},
language = {en},
number = {4},
urldate = {2022-09-20},
journal = {Journal of Applied Physics},
author = {Caldwell, Joshua D. and Stahlbush, Robert E. and Ancona, Mario G. and Glembocki, Orest J. and Hobart, Karl D.},
month = aug,
year = {2010},
pages = {044503},
file = {Caldwell 等。 - 2010 - On the driving force for recombination-induced sta.pdf:/home/chn/Zotero/storage/BRT4UJBM/Caldwell 等。 - 2010 - On the driving force for recombination-induced sta.pdf:application/pdf},
}
@article{iwahashi_extension_2017,
title = {Extension of {Stacking} {Faults} in {4H}-{SiC} pn {Diodes} under a {High} {Current} {Pulse} {Stress}},
volume = {897},
issn = {1662-9752},
url = {https://www.scientific.net/MSF.897.218},
doi = {10.4028/www.scientific.net/MSF.897.218},
abstract = {We investigated the expansion of stacking faults (SFs) under a high current pulse stress in detail. In situ observations showed bar-shaped SFs and two types of triangle SFs with different nucleation sites. The calculated partial dislocation velocity of the bar-shaped SFs was four times faster than that of the triangle SFs. The temperature dependence of the partial dislocation velocity was used to estimate activation energies of 0.23±0.02 eV for bar-shaped SFs and 0.27±0.05 eV for triangle SFs. We also compared the electrical characteristics before and after the stress. The forward voltage drop slightly increased by 0.05 V, and the leakage current did not increase.},
language = {en},
urldate = {2022-09-22},
journal = {Materials Science Forum},
author = {Iwahashi, Yohei and Miyazato, Masaki and Miyajima, Masaaki and Yonezawa, Yoshiyuki and Kato, Tomohisa and Fujiwara, Hirokazu and Hamada, Kimimori and Otsuki, Akihiro and Okumura, Hajime},
month = may,
year = {2017},
pages = {218--221},
file = {Iwahashi 等。 - 2017 - Extension of Stacking Faults in 4H-SiC pn Diodes u.pdf:/home/chn/Zotero/storage/8N5CPXAY/Iwahashi 等。 - 2017 - Extension of Stacking Faults in 4H-SiC pn Diodes u.pdf:application/pdf},
}
@article{okada_dependences_2018,
title = {Dependences of contraction/expansion of stacking faults on temperature and current density in {4H}-{SiC} p-i-n diodes},
volume = {57},
issn = {0021-4922, 1347-4065},
url = {https://iopscience.iop.org/article/10.7567/JJAP.57.061301},
doi = {10.7567/JJAP.57.061301},
language = {en},
number = {6},
urldate = {2022-09-22},
journal = {Japanese Journal of Applied Physics},
author = {Okada, Aoi and Nishio, Johji and Iijima, Ryosuke and Ota, Chiharu and Goryu, Akihiro and Miyazato, Masaki and Ryo, Mina and Shinohe, Takashi and Miyajima, Masaaki and Kato, Tomohisa and Yonezawa, Yoshiyuki and Okumura, Hajime},
month = jun,
year = {2018},
pages = {061301},
file = {Okada 等。 - 2018 - Dependences of contractionexpansion of stacking f.pdf:/home/chn/Zotero/storage/ZDPSUFFI/Okada 等。 - 2018 - Dependences of contractionexpansion of stacking f.pdf:application/pdf},
}
@article{iijima_correlation_2017,
title = {Correlation between shapes of {Shockley} stacking faults and structures of basal plane dislocations in {4H}-{SiC} epilayers},
volume = {97},
issn = {1478-6435, 1478-6443},
url = {https://www.tandfonline.com/doi/full/10.1080/14786435.2017.1350788},
doi = {10.1080/14786435.2017.1350788},
abstract = {Shockley-type stacking faults expanded in 4H–SiC epilayers induced by ultraviolet illumination were investigated using a photoluminescence imaging method, a photoluminescence mapping method and X-ray topography. After ultraviolet illumination, more than 30 patterns of Shockley-type stacking faults which expanded from perfect basal plane dislocations were observed by photoluminescence imaging. The initial basal plane dislocations were crystallographically classified, and individual shapes of expanded Shockley-type stacking faults were predicted. The correspondence between the predicted shapes and observed ones was discussed.},
language = {en},
number = {30},
urldate = {2022-09-24},
journal = {Philosophical Magazine},
author = {Iijima, Akifumi and Kamata, Isaho and Tsuchida, Hidekazu and Suda, Jun and Kimoto, Tsunenobu},
month = oct,
year = {2017},
pages = {2736--2752},
file = {Iijima 等。 - 2017 - Correlation between shapes of Shockley stacking fa.pdf:/home/chn/Zotero/storage/2G3ZSIAB/Iijima 等。 - 2017 - Correlation between shapes of Shockley stacking fa.pdf:application/pdf},
}
@article{mahadik_ultraviolet_2012,
title = {Ultraviolet {Photoluminescence} {Imaging} of {Stacking} {Fault} {Contraction} in {4H}-{SiC} {Epitaxial} {Layers}},
volume = {717-720},
issn = {1662-9752},
url = {https://www.scientific.net/MSF.717-720.391},
doi = {10.4028/www.scientific.net/MSF.717-720.391},
abstract = {Shockley stacking fault (SSF) contraction in 4H-SiC was investigated, in-situ, under varying temperature and ultraviolet (UV) intensity. Contraction of single SSFs at room temperature was observed for the first time under low power UV excitation of 0.04 W/cm2. At temperatures above 150 oC, complete SSF contraction occurred for UV power at 0.2 W/cm2. In contrast to expansion, SSF contraction occurred in discrete jumps between pinning sites along existing C-core partials. Luminescence from the pinning sites suggest they may be local concentrations of point defects. Additionally, a change in the line direction of the Si-core partials by {\textasciitilde}25o off the {\textless}112¯ 0{\textgreater} direction was observed.},
language = {en},
urldate = {2022-09-24},
journal = {Materials Science Forum},
author = {Mahadik, Nadeemullah A. and Stahlbush, Robert E. and Caldwell, Joshua D. and Hobart, Karl D.},
month = may,
year = {2012},
pages = {391--394},
file = {Mahadik 等。 - 2012 - Ultraviolet Photoluminescence Imaging of Stacking .pdf:/home/chn/Zotero/storage/MYMBL92A/Mahadik 等。 - 2012 - Ultraviolet Photoluminescence Imaging of Stacking .pdf:application/pdf},
}
@article{miyanagi_annealing_2006,
title = {Annealing effects on single {Shockley} faults in {4H}-{SiC}},
volume = {89},
issn = {0003-6951, 1077-3118},
url = {http://aip.scitation.org/doi/10.1063/1.2234740},
doi = {10.1063/1.2234740},
language = {en},
number = {6},
urldate = {2022-09-24},
journal = {Applied Physics Letters},
author = {Miyanagi, Toshiyuki and Tsuchida, Hidekazu and Kamata, Isaho and Nakamura, Tomonori and Nakayama, Koji and Ishii, Ryousuke and Sugawara, Yoshitaka},
month = aug,
year = {2006},
pages = {062104},
file = {Miyanagi 等。 - 2006 - Annealing effects on single Shockley faults in 4H-.pdf:/home/chn/Zotero/storage/33N8FRKW/Miyanagi 等。 - 2006 - Annealing effects on single Shockley faults in 4H-.pdf:application/pdf},
}
@article{nishio_triangular_2020,
title = {Triangular {Single} {Shockley} {Stacking} {Fault} {Analyses} on {4H}-{SiC} {PiN} {Diode} with {Forward} {Voltage} {Degradation}},
volume = {49},
issn = {0361-5235, 1543-186X},
url = {https://link.springer.com/10.1007/s11664-020-08133-7},
doi = {10.1007/s11664-020-08133-7},
language = {en},
number = {9},
urldate = {2022-09-24},
journal = {Journal of Electronic Materials},
author = {Nishio, Johji and Okada, Aoi and Ota, Chiharu and Kushibe, Mitsuhiro},
month = sep,
year = {2020},
pages = {5232--5239},
file = {Nishio 等。 - 2020 - Triangular Single Shockley Stacking Fault Analyses.pdf:/home/chn/Zotero/storage/SPG9HXBM/Nishio 等。 - 2020 - Triangular Single Shockley Stacking Fault Analyses.pdf:application/pdf},
}
@article{okumura_present_2006,
title = {Present {Status} and {Future} {Prospect} of {Widegap} {Semiconductor} {High}-{Power} {Devices}},
volume = {45},
issn = {0021-4922, 1347-4065},
doi = {10.1143/JJAP.45.7565},
language = {en},
number = {10A},
urldate = {2022-09-27},
journal = {Japanese Journal of Applied Physics},
@@ -18,6 +156,7 @@
title = {Status of {Silicon} {Carbide} ({SiC}) as a {Wide}-bandgap {Emiconductor} for {High}-temperature {Applications}: a {Review}},
volume = {39},
doi = {10.1016/0038-1101(96)00045-7},
language = {en},
number = {10},
journal = {Solid-State Electronics},
author = {Casady, J B and Johnson, R W},
@@ -33,6 +172,7 @@
issn = {13698001},
doi = {10.1016/j.mssp.2017.11.003},
abstract = {This paper reports recent advances in high-quality 4H-SiC epitaxial growth. The modern 4H-SiC epitaxial reactors, techniques to improve growth rates and large-diameter uniformity and reduce defect densities are discussed. A single-wafer vertical-type epitaxial reactor is newly developed and employed to grow 150 mm-diameter 4H-SiC epilayers. Using the reactor, high-speed wafer rotation is confirmed effective, both for enhancing growth rates and improving thickness and doping uniformities. Current levels of reducing particle-induced defects, in-grown stacking faults and basal plane dislocations and controlling carrier lifetimes are also reviewed.},
language = {en},
urldate = {2022-10-06},
journal = {Materials Science in Semiconductor Processing},
author = {Tsuchida, Hidekazu and Kamata, Isaho and Miyazawa, Tetsuya and Ito, Masahiko and Zhang, Xuan and Nagano, Masahiro},
@@ -50,6 +190,7 @@
url = {https://linkinghub.elsevier.com/retrieve/pii/S0169433222024771},
doi = {10.1016/j.apsusc.2022.154949},
abstract = {Silicon carbide (SiC) has gained increased interest due to industry demand, especially for the 4H-SiC. Never­ theless, the ‘structural mutation’ in the 4H-SiC epitaxy is in urgent need of investigation and proper solution as the epitaxial thickness/wafer size increases. In this study, growth monomers in the step-flow mode were firstly investigated by the first-principles calculations for their dynamic and kinetic behaviours from an atomic level. The stability (by the comprehensive analyses of total energies, chemical potentials, and formation enthalpies) and the location of adsorptions were studied to reveal the dynamics. Meanwhile, the potential barrier of Si-Si interaction and phonon spectra were determined to understand the kinetics. We found monomers could be selected by controlling chemical potentials to make ordering growth. Secondly, two methods were thus inferred to select monomers to adsorb on atomic step surfaces in an orderly fashion and were verified in a six-inch epitaxy. Thirdly, a protocol was designed to restrict the extension of basal plane dislocation (BPD) from sub­ strates, a reduction greater than five orders of magnitude was gained but without time compromise in the thickfilm epitaxy. This study provided new insights into growth on the 4H-SiC (0001) atomic step surfaces and a new way of 4H-SiC homo-epitaxy.},
language = {en},
urldate = {2022-10-06},
journal = {Applied Surface Science},
author = {Sun, Yongqiang and Kang, Wenyu and Chen, Haonan and Chen, Xinlu and Dong, Yue and Lin, Wei and Kang, Junyong},
@@ -59,6 +200,23 @@
file = {Sun et al. - 2022 - Selection of growth monomers on the 4H-SiC (0001) .pdf:/home/chn/Zotero/storage/VTGL4G53/Sun et al. - 2022 - Selection of growth monomers on the 4H-SiC (0001) .pdf:application/pdf},
}
@article{harada_suppression_2022,
title = {Suppression of stacking fault expansion in a {4H}-{SiC} epitaxial layer by proton irradiation},
volume = {12},
issn = {2045-2322},
doi = {10.1038/s41598-022-17060-y},
abstract = {SiC bipolar degradation, which is caused by stacking fault expansion from basal plane dislocations in a SiC epitaxial layer or near the interface between the epitaxial layer and the substrate, is one of the critical problems inhibiting widespread usage of high-voltage SiC bipolar devices. In the present study, we investigated the stacking fault expansion behavior under UV illumination in a 4H-SiC epitaxial layer subjected to proton irradiation. X-ray topography observations revealed that proton irradiation suppressed stacking fault expansion. Excess carrier lifetime measurements showed that stacking fault expansion was suppressed in 4H-SiC epitaxial layers with proton irradiation at a fluence of 1 × 1011cm−2without evident reduction of the excess carrier lifetime. Furthermore, stacking fault expansion was also suppressed even after high-temperature annealing to recover the excess carrier lifetime. These results implied that passivation of dislocation cores by protons hinders recombination-enhanced dislocation glide motion under UV illumination.},
language = {en},
number = {1},
urldate = {2022-10-06},
journal = {Scientific Reports},
author = {Harada, Shunta and Mii, Toshiki and Sakane, Hitoshi and Kato, Masashi},
month = aug,
year = {2022},
pages = {13542},
file = {Harada et al. - 2022 - Suppression of stacking fault expansion in a 4H-Si.pdf:/home/chn/Zotero/storage/VJ7H4G59/Harada et al. - 2022 - Suppression of stacking fault expansion in a 4H-Si.pdf:application/pdf},
}
@article{harada_observation_2022,
title = {Observation of in-plane shear stress fields in off-axis {SiC} wafers by birefringence imaging},
volume = {55},
@@ -66,6 +224,7 @@
url = {https://scripts.iucr.org/cgi-bin/paper?S1600576722006483},
doi = {10.1107/S1600576722006483},
abstract = {For the nondestructive characterization of SiC wafers for power device application, birefringence imaging is one of the promising methods. In the present study, it is demonstrated that birefringence image contrast variation in off-axis SiC wafers corresponds to the in-plane shear stress under conditions slightly deviating from crossed Nicols according to both theoretical consideration and experimental observation. The current results indicate that the characterization of defects in SiC wafers is possible to achieve by birefringence imaging.},
language = {en},
number = {4},
urldate = {2023-06-14},
journal = {Journal of Applied Crystallography},
@@ -75,3 +234,308 @@
pages = {1029--1032},
file = {Harada and Murayama - 2022 - Observation of in-plane shear stress fields in off.pdf:/home/chn/Zotero/storage/X6LVEJRU/Harada and Murayama - 2022 - Observation of in-plane shear stress fields in off.pdf:application/pdf},
}
@article{kudriavtsev_quantitative_2003,
title = {Quantitative {SIMS} analysis of {SiC}},
volume = {35},
copyright = {http://onlinelibrary.wiley.com/termsAndConditions\#vor},
issn = {0142-2421, 1096-9918},
url = {https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/10.1002/sia.1561},
doi = {10.1002/sia.1561},
abstract = {Abstract
We performed a systematic study of ion‐implanted 6H‐SiC standards to find the optimal regimes for SIMS analysis. Relative sensitivity factors (RSFs) were acquired for operating conditions typical of practical SIMS applications. The experimental SiC RSFs were compared with those found for silicon:
1
the matrix effect was insignificant in most cases. It was found that the SiO
−
cluster ion cannot represent correctly the real oxygen distribution in SiC. The physics of the effect is discussed. Copyright © 2003 John Wiley \& Sons, Ltd.},
language = {en},
number = {6},
urldate = {2025-06-03},
journal = {Surface and Interface Analysis},
author = {Kudriavtsev, Yu. and Villegas, A. and Godines, A. and Asomoza, R. and Usov, I.},
month = jun,
year = {2003},
pages = {491--495},
file = {PDF:/home/chn/Zotero/storage/GNZIFH2D/Kudriavtsev et al. - 2003 - Quantitative SIMS analysis of SiC.pdf:application/pdf},
}
@article{kim_characteristics_2024,
title = {Characteristics of the {Discoloration} {Switching} {Phenomenon} of {4H}-{SiC} {Single} {Crystals} {Grown} by {PVT} {Method} {Using} {ToF}-{SIMS} and {Micro}-{Raman} {Analysis}},
volume = {17},
copyright = {https://creativecommons.org/licenses/by/4.0/},
issn = {1996-1944},
url = {https://www.mdpi.com/1996-1944/17/5/1005},
doi = {10.3390/ma17051005},
abstract = {The discoloration switching appearing in the initial and final growth stages of 4H-silicon carbide (4H-SiC) single crystals grown using the physical vapor transport (PVT) technique was investigated. This phenomenon was studied, investigating the correlation with linear-type micropipe defects on the surface of 4H-SiC single crystals. Based on the experimental results obtained using time-of-flight secondary ion mass spectrometry (ToF-SIMS) and micro-Raman analysis, it was deduced that the orientation of the 4H-SiC c-axis causes an axial change that correlates with low levels of carbon. In addition, it was confirmed that the incorporation of additional elements and the concentrations of these doped impurity elements were the main causes of discoloration and changes in growth orientation. Overall, this work provides guidelines for evaluating the discoloration switching in 4H-SiC single crystals and contributes to a greater understanding of this phenomenon.},
language = {en},
number = {5},
urldate = {2025-06-03},
journal = {Materials},
author = {Kim, Seul-Ki and Kim, Hajun and Kim, Hyun Sik and Hong, Tae Eun and Lee, Younki and Jung, Eun Young},
month = feb,
year = {2024},
pages = {1005},
file = {PDF:/home/chn/Zotero/storage/JXDKH48U/Kim et al. - 2024 - Characteristics of the Discoloration Switching Phenomenon of 4H-SiC Single Crystals Grown by PVT Met.pdf:application/pdf},
}
@article{noguchi_comparative_2021,
title = {Comparative {Study} of {Hall} {Effect} {Mobility} in {Inversion} {Layer} of {4H}-{SiC} {MOSFETs} {With} {Nitrided} and {Phosphorus}-{Doped} {Gate} {Oxides}},
volume = {68},
copyright = {https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html},
issn = {0018-9383, 1557-9646},
url = {https://ieeexplore.ieee.org/document/9632361/},
doi = {10.1109/TED.2021.3125284},
abstract = {In this study, the inversion layer mobility characteristics in Si-face 4H silicon carbide (SiC) metaloxide-semiconductor field-effect transistors (MOSFETs) with nitrided and phosphorus-doped gate oxides were compared using Hall effect measurements. The inversion layer mobility was evaluated by applying a body bias and changing the temperature. The carrier scattering properties were determined for elevated temperatures (i.e., 473 K), at which point the impact of Coulomb scattering decreases and that of phonon scattering increases. The phonon-limited mobility of these MOSFETs was almost the same when plotted as a function of the effective normal electric field in the inversion layer, possibly representing the nature of the thermally grown SiO2/SiC interface. On the basis of this finding, the effect of phonon scattering was separated from the inversion layer mobility. The MOSFETs exhibited a remarkable difference in Coulomb scattering: the MOSFETs with phosphorus-doped gate oxide exhibited a more rapid increase in Coulomb-limited mobility with increasing surface carrier density than did the MOSFETs with nitride gate oxide. This resulted from the effective suppression of Coulomb scattering in the inversion layer, which is one of the reasons why phosphorus-doped gate oxide achieves higher inversion layer mobility than nitrided gate oxide. These results show that the inversion layer mobility of SiC MOSFETs can be modeled using a conventional framework of phonon, Coulomb, and surface roughness scattering. Therefore, the suppression of Coulomb scattering is key to further improving the inversion layer mobility of SiC MOSFETs.},
language = {en},
number = {12},
urldate = {2025-06-03},
journal = {IEEE Transactions on Electron Devices},
author = {Noguchi, Munetaka and Watanabe, Tomokatsu and Watanabe, Hiroshi and Kita, Koji and Miura, Naruhisa},
month = dec,
year = {2021},
pages = {6321--6329},
file = {PDF:/home/chn/Zotero/storage/WGL6FA4D/Noguchi et al. - 2021 - Comparative Study of Hall Effect Mobility in Inversion Layer of 4H-SiC MOSFETs With Nitrided and Pho.pdf:application/pdf},
}
@article{asada_hall_2016,
title = {Hall scattering factors in p-type {4H}-{SiC} with various doping concentrations},
volume = {9},
issn = {1882-0778, 1882-0786},
url = {https://iopscience.iop.org/article/10.7567/APEX.9.041301},
doi = {10.7567/APEX.9.041301},
abstract = {Abstract The Hall scattering factor (γ H ) in p-type 4H-SiC with various aluminum doping concentrations of 5.8 × 10 14 –7.1 × 10 18 cm −3 was investigated from 300 to 900 K. γ H was determined by comparing the Hall coefficient with the theoretical carrier concentration derived from acceptor and donor concentrations obtained from secondary ion mass spectrometry and capacitance–voltage measurements. γ H decreased with increasing temperature or doping concentration; γ H = 1–0.4 for the doping concentration of 5.8 × 10 14 cm −3 and γ H = 0.5–0.2 for the doping concentration of 7.1 × 10 18 cm −3 . The dependence might be caused by the anisotropic and nonparabolic valence band structure of 4H-SiC.},
language = {en},
number = {4},
urldate = {2025-06-03},
journal = {Applied Physics Express},
author = {Asada, Satoshi and Okuda, Takafumi and Kimoto, Tsunenobu and Suda, Jun},
month = apr,
year = {2016},
pages = {041301},
file = {PDF:/home/chn/Zotero/storage/AXK4NKG7/Asada et al. - 2016 - Hall scattering factors in p-type 4H-SiC with various doping concentrations.pdf:application/pdf},
}
@article{egoavil_atomic_2014,
title = {Atomic resolution mapping of phonon excitations in {STEM}-{EELS} experiments},
volume = {147},
issn = {03043991},
url = {https://linkinghub.elsevier.com/retrieve/pii/S0304399114000904},
doi = {10.1016/j.ultramic.2014.04.011},
abstract = {Atomically resolved electron energy-loss spectroscopy experiments are commonplace in modern aberration-corrected transmission electron microscopes. Energy resolution has also been increasing steadily with the continuous improvement of electron monochromators. Electronic excitations however are known to be delocalized due to the long range interaction of the charged accelerated electrons with the electrons in a sample. This has made several scientists question the value of combined high spatial and energy resolution for mapping interband transitions and possibly phonon excitation in crystals. In this paper we demonstrate experimentally that atomic resolution information is indeed available at very low energy losses around 100 meV expressed as a modulation of the broadening of the zero loss peak. Careful data analysis allows us to get a glimpse of what are likely phonon excitations with both an energy loss and gain part. These experiments confirm recent theoretical predictions on the strong localization of phonon excitations as opposed to electronic excitations and show that a combination of atomic resolution and recent developments in increased energy resolution will offer great benefit for mapping phonon modes in real space.},
language = {en},
urldate = {2025-06-03},
journal = {Ultramicroscopy},
author = {Egoavil, R. and Gauquelin, N. and Martinez, G.T. and Van Aert, S. and Van Tendeloo, G. and Verbeeck, J.},
month = dec,
year = {2014},
pages = {1--7},
file = {PDF:/home/chn/Zotero/storage/5YTDSVNA/Egoavil et al. - 2014 - Atomic resolution mapping of phonon excitations in STEM-EELS experiments.pdf:application/pdf},
}
@article{tong_temperature-dependent_2018,
title = {Temperature-dependent infrared optical properties of {3C}-, {4H}- and {6H}-{SiC}},
volume = {537},
issn = {09214526},
url = {https://linkinghub.elsevier.com/retrieve/pii/S092145261830142X},
doi = {10.1016/j.physb.2018.02.023},
abstract = {The temperature-dependent optical properties of cubic (3C) and hexagonal (4H and 6H) silicon carbide are investigated in the infrared range of 2–16 μm both by experimental measurements and numerical simulations. The temperature in experimental measurement is up to 593 K, while the numerical method can predict the optical properties at elevated temperatures. To investigate the temperature effect, the temperature-dependent damping parameter in the Lorentz model is calculated based on anharmonic lattice dynamics method, in which the harmonic and anharmonic interatomic force constants are determined from first-principles calculations. The infrared phonon modes of silicon carbide are determined from first-principles calculations. Based on first-principles calculations, the Lorentz model is parameterized without any experimental fitting data and the temperature effect is considered. In our investigations, we find that the increasing temperature induces a small reduction of the reflectivity in the range of 10–13 μm. More importantly, it also shows that our first-principles calculations can predict the infrared optical properties at high-temperature effectively which is not easy to be obtained through experimental measurements.},
language = {en},
urldate = {2025-06-03},
journal = {Physica B: Condensed Matter},
author = {Tong, Zhen and Liu, Linhua and Li, Liangsheng and Bao, Hua},
month = may,
year = {2018},
pages = {194--201},
file = {PDF:/home/chn/Zotero/storage/RISQQUQJ/Tong et al. - 2018 - Temperature-dependent infrared optical properties of 3C-, 4H- and 6H-SiC.pdf:application/pdf},
}
@article{pluchery_infrared_2012,
title = {Infrared spectroscopy characterization of {3C}–{SiC} epitaxial layers on silicon},
volume = {45},
issn = {0022-3727, 1361-6463},
url = {https://iopscience.iop.org/article/10.1088/0022-3727/45/49/495101},
doi = {10.1088/0022-3727/45/49/495101},
abstract = {We have measured the transmission Fourier transform infrared spectra of cubic silicon carbide (3C–SiC polytype) epitaxial layer with a 20 µm thickness on a 200 µm thick silicon substrate. Spectra were recorded in the 400–4000 cm−1 wavenumber range. A novel approach of IR spectra computations based on the recursion capability of the C programming language is presented on the basis of polarized light propagation in layered media using generalized Fresnel’s equations. The complex refractive indices are the only input parameters. A remarkable agreement is found between all of the experimental SiC and Si spectral features and the calculated spectra. A comprehensive assignment of (i) the two fundamental transverse optical (TO) (790 cm−1) and longitudinal optical (LO) (970 cm−1) phonon modes of 3C–SiC, (ii) with their overtones (1522–1627 cm−1) and (iii) the two-phonon optical-acoustical summation bands (1311–1409 cm−1) is achieved on the basis of available literature data. This approach allows sorting out the respective contributions of the Si substrate and SiC upper layer. Such calculations can be applied to any medium, provided that the complex refractive index data are known.},
language = {en},
number = {49},
urldate = {2025-06-03},
journal = {Journal of Physics D: Applied Physics},
author = {Pluchery, Olivier and Costantini, Jean-Marc},
month = dec,
year = {2012},
pages = {495101},
file = {PDF:/home/chn/Zotero/storage/MV3Y48T4/Pluchery and Costantini - 2012 - Infrared spectroscopy characterization of 3C–SiC epitaxial layers on silicon.pdf:application/pdf},
}
@article{feldman_phonon_1968,
title = {Phonon {Dispersion} {Curves} by {Raman} {Scattering} in {SiC}, {Polytypes} 3 {C} , 4 {H} , 6 {H} , 1 5 {R} , and 2 1 {R}},
volume = {173},
copyright = {http://link.aps.org/licenses/aps-default-license},
issn = {0031-899X},
url = {https://link.aps.org/doi/10.1103/PhysRev.173.787},
doi = {10.1103/PhysRev.173.787},
language = {en},
number = {3},
urldate = {2025-06-03},
journal = {Physical Review},
author = {Feldman, D. W. and Parker, James H. and Choyke, W. J. and Patrick, Lyle},
month = sep,
year = {1968},
pages = {787--793},
file = {PDF:/home/chn/Zotero/storage/AYLRJSTD/Feldman et al. - 1968 - Phonon Dispersion Curves by Raman Scattering in SiC, Polytypes 3 C , 4 H , 6 H , 1 5 R , and 2 1 R.pdf:application/pdf},
}
@article{nakashima_raman_2013,
title = {Raman intensity profiles of zone-folded modes in {SiC}: {Identification} of stacking sequence of {10H}-{SiC}},
volume = {114},
issn = {0021-8979, 1089-7550},
shorttitle = {Raman intensity profiles of zone-folded modes in {SiC}},
url = {https://pubs.aip.org/jap/article/114/19/193510/992087/Raman-intensity-profiles-of-zone-folded-modes-in},
doi = {10.1063/1.4828996},
abstract = {Raman intensity profiles are measured for 10H-SiC crystals, for which various zone-folded phonon modes are observed. Raman intensity profiles are calculated based on a bond polarizability model assuming several stacking sequences for the 10H polytype using a linear chain model. Among several candidates for the stacking sequences, the 3322 stacking structure provides the best-fit profile for experimental spectral profiles. The hexagonality value of 0.4 predicted from the stacking sequence of this polytype is consistent with that derived from the frequency splitting between the experimental A1 and E-type transverse optical modes. This fact is consistent with an empirical rule that the value of the reduced wavevector for the strongest folded transverse acoustic and optical modes are equal to the hexagonality of the polytype. In the present analysis of the Raman intensity profiles, the calculated intensity profiles for specified folded transverse optical modes are found to be relatively strong and strikingly dependent on force-field parameters in α-SiC that consists of the mixture of the cubic and hexagonal stacking structures. These force-field parameters can reproduce well the experimental Raman intensity profiles of various SiC polytypes including 10H-SiC.},
language = {en},
number = {19},
urldate = {2025-06-03},
journal = {Journal of Applied Physics},
author = {Nakashima, S. and Tomita, T. and Kuwahara, N. and Mitani, T. and Tomobe, M. and Nishizawa, S. and Okumura, H.},
month = nov,
year = {2013},
pages = {193510},
file = {PDF:/home/chn/Zotero/storage/PAGC4NNX/Nakashima et al. - 2013 - Raman intensity profiles of zone-folded modes in SiC Identification of stacking sequence of 10H-SiC.pdf:application/pdf},
}
@article{guo_characterization_2012,
title = {Characterization of {Polytype} {Distributions} in {Nitrogen}-doped {6H}-{SiC} {Single} {Crystal} by {Raman} {Mapping}: {Characterization} of {Polytype} {Distributions} in {Nitrogen}-doped {6H}-{SiC} {Single} {Crystal} by {Raman} {Mapping}},
volume = {27},
issn = {1000-324X},
shorttitle = {Characterization of {Polytype} {Distributions} in {Nitrogen}-doped {6H}-{SiC} {Single} {Crystal} by {Raman} {Mapping}},
url = {http://pub.chinasciencejournal.com/article/getArticleRedirect.action?doiCode=10.3724/SP.J.1077.2012.00609},
doi = {10.3724/SP.J.1077.2012.00609},
abstract = {Nitrogen-doped 6H-SiC crystal with the diameter of 2-inch was grown along [0001] direction by physical vapor transport method. The spatial distribution of different polytypes such as 6H-SiC, 4H-SiC and 15R-SiC was characterized by mapping Raman spectra. The formation and evolution of different polytypes were investigated during the growth progress. 15R-SiC and 4H-SiC were observed in the as-grown 6H-SiC single crystal. Two different polytype regions are observed from the spatial distribution of different polytypes. One region originates from the growth interface of different polytypes. This region has higher nitrogen doping level and carrier concentration, and the area can become large during the growth process. The other region is dominated by 15R-SiC which appears in the main 6H-SiC due to the perturbation in growth temperature, pressure, etc. This region has less effect on the crystal quality, which could be inhibited by increasing the growth temperature.},
language = {zh},
number = {6},
urldate = {2025-06-03},
journal = {Journal of Inorganic Materials},
author = {Guo, Xiao and Liu, Xue-Chao and Xin, Jun and Yang, Jian-Hua and Shi, Er-Wei},
month = aug,
year = {2012},
pages = {609--614},
file = {PDF:/home/chn/Zotero/storage/UD55XBUQ/Guo et al. - 2012 - Characterization of Polytype Distributions in Nitrogen-doped 6H-SiC Single Crystal by Raman Mapping.pdf:application/pdf},
}
@inproceedings{yan_study_2016,
address = {Beijing},
title = {Study of morphology defects in {4H}-{SiC} thick epitaxial layers grown on 4° off-axis {Si}-face substrates},
isbn = {978-1-5090-4613-3},
url = {https://ieeexplore.ieee.org/document/7803743/},
doi = {10.1109/IFWS.2016.7803743},
abstract = {The crystallographic structure and origins of morphology defects observed in 4°off-axis Si-face thick 4H-SiC epitaxial layers were investigated by Nomarski microscope and Raman spectroscopy. The growth direction of these morphology defects is consistent with the step-flow direction, all of the defect include a certain core, which indicates that the defects were originated from certain cores. These cores of the morphology defects contain 3C poly-crystalline grains based on the Raman spectroscopy characterization. The head part of the defect formed during epitaxial layers growth and their formation is attributed to the foreign particles. The formation mechanisms of these obtuse morphology defects are discussed based on our model. It can be concluded that foreign particles fall down on the surface during the 4H-SiC epitaxy that disturb the normal step flow mode and lead to the 3C-SiC nucleation, which is the origination of the morphology defects.},
language = {en},
urldate = {2025-06-03},
booktitle = {2016 13th {China} {International} {Forum} on {Solid} {State} {Lighting}: {International} {Forum} on {Wide} {Bandgap} {Semiconductors} {China} ({SSLChina}: {IFWS})},
publisher = {IEEE},
author = {Yan, Guoguo and Zhang, Feng and Liu, Xingfang and Wang, Lei and Zhao, Wanshun and Sun, Guosheng and Zeng Key, Yiping},
month = nov,
year = {2016},
pages = {6--10},
file = {PDF:/home/chn/Zotero/storage/82RJ3M4G/Yan et al. - 2016 - Study of morphology defects in 4H-SiC thick epitaxial layers grown on 4° off-axis Si-face substrates.pdf:application/pdf},
}
@article{hundhausen_characterization_2008,
title = {Characterization of defects in silicon carbide by {Raman} spectroscopy},
volume = {245},
copyright = {http://onlinelibrary.wiley.com/termsAndConditions\#vor},
issn = {0370-1972, 1521-3951},
url = {https://onlinelibrary.wiley.com/doi/10.1002/pssb.200844052},
doi = {10.1002/pssb.200844052},
abstract = {Abstract
We demonstrate the application of Raman spectroscopy as an optical non‐contact method for the characterization of silicon carbide (SiC). The Raman spectra provide information about the polytype and thus can give direct information about microscopic inclusions of hexagonal polytypes in 3C‐SiC grown by chemical vapor deposition (CVD) after annealing at elevated temperatures. Polytype conversion sets in at a about 1700 °C and at higher temperatures eventually results in larger domains of 6H‐SiC where twin boundaries act as barriers against a complete polytype conversion. We study shallow donor states of phosphorus‐ and nitrogen‐doped SiC using low temperature electronic Raman spectroscopy. The various low frequency transitions observed in nitrogen doped SiC are assigned to the valley‐orbit transitions of electrons in the 1s‐ground states of donors that occupy inequivalent lattice sites. During vacuum annealing at elevated temperature graphitization of the SiC surface occurs. Raman spectroscopy is used to verify that under well controlled conditions a monoatomic graphene layer exists. We observe a phonon hardening of that layer compared to free standing graphene that we ascribe mainly to strain induced by different thermal expansion coefficients of graphite and SiC. (© 2008 WILEY‐VCH Verlag GmbH \& Co. KGaA, Weinheim)},
language = {en},
number = {7},
urldate = {2025-06-03},
journal = {physica status solidi (b)},
author = {Hundhausen, M. and Püsche, R. and Röhrl, J. and Ley, L.},
month = jul,
year = {2008},
pages = {1356--1368},
file = {PDF:/home/chn/Zotero/storage/7SFXQFGR/Hundhausen et al. - 2008 - Characterization of defects in silicon carbide by Raman spectroscopy.pdf:application/pdf},
}
@article{song_depth_2020,
title = {Depth {Profiling} of {Ion}-{Implanted} {4H}–{SiC} {Using} {Confocal} {Raman} {Spectroscopy}},
volume = {10},
copyright = {https://creativecommons.org/licenses/by/4.0/},
issn = {2073-4352},
url = {https://www.mdpi.com/2073-4352/10/2/131},
doi = {10.3390/cryst10020131},
abstract = {For silicon carbide (SiC) processed by ion-implantation, dedicated test structure fabrication or destructive sample processing on test wafers are usually required to obtain depth profiles of electrical characteristics such as carrier concentration. In this study, a rapid and non-destructive approach for depth profiling is presented that uses confocal Raman microscopy. As an example, a 4H–SiC substrate with an epitaxial layer of several micrometers thick and top layer in nanoscale that was modified by ion-implantation was characterized. From the Raman depth profiling, longitudinal optical (LO) mode from the epitaxial layer and longitudinal optical phonon-plasmon coupled (LOPC) mode from the substrate layer can be sensitively distinguished at the interface. The position profile of the LOPC peak intensity in the depth direction was found to be effective in estimating the thickness of the epitaxial layer. For three kinds of epitaxial layer with thicknesses of 5.3 µm, 6 µm, and 7.5 µm, the average deviations of the Raman depth analysis were −1.7 µm, −1.2 µm, and −1.4 µm, respectively. Moreover, when moving the focal plane from the heavily doped sample ({\textasciitilde}1018 cm−3) to the epitaxial layer ({\textasciitilde}1016 cm−3), the LOPC peak showed a blue shift. The twice travel of the photon (excitation and collection) through the ion-implanted layer with doping concentrations higher than 1 × 1018 cm−3 led to a difference in the LOPC peak position for samples with the same epitaxial layer and substrate layer. Furthermore, the influences of the setup in terms of pinhole size and numerical aperture of objective lens on the depth profiling results were studied. Different from other research on Raman depth profiling, the 50× long working distance objective lens (50L× lens) was found more suitable than the 100× lens for the depth analysis 4H–SiC with a multi-layer structure.},
language = {en},
number = {2},
urldate = {2025-06-04},
journal = {Crystals},
author = {Song, Ying and Xu, Zongwei and Liu, Tao and Rommel, Mathias and Wang, Hong and Wang, Yufang and Fang, Fengzhou},
month = feb,
year = {2020},
pages = {131},
file = {PDF:/home/chn/Zotero/storage/PR6FW4VH/Song et al. - 2020 - Depth Profiling of Ion-Implanted 4H–SiC Using Confocal Raman Spectroscopy.pdf:application/pdf},
}
@article{harima_raman_1995,
title = {Raman scattering from anisotropic {LO}-phonon–plasmon–coupled mode in \textit{n} -type {4H}– and {6H}–{SiC}},
volume = {78},
issn = {0021-8979, 1089-7550},
url = {https://pubs.aip.org/jap/article/78/3/1996/489713/Raman-scattering-from-anisotropic-LO-phonon},
doi = {10.1063/1.360174},
abstract = {LO-phonon–plasmon–coupled modes in n-type 4H– and 6H–SiC single crystals with free-carrier concentrations of 1016–1018 cm−3 have been measured by Raman scattering at room temperature. The axial-type mode for which plasma oscillation and atomic displacement are parallel to the c axis, and the planar-type mode for which these oscillations lie in the c plane, have been individually observed. From a line-shape analysis of the observed spectra, the plasmon frequency, carrier damping, and phonon damping have been deduced. These quantities have large differences between the axial- and planar-type mode in 6H–SiC, indicating its large crystal anisotropy. On the contrary, 4H–SiC shows small anisotropy. The longitudinal and transverse effective mass components of the electron have been determined from the plasmon frequency using carrier densities derived from Hall measurements. The deduced values are m∥=1.4m0 and m⊥=0.35m0 for 6H–SiC, and m∥=0.48m0 and m⊥=0.30m0 for 4H–SiC. The carrier mobility obtained from the analysis is also anisotropic. This is consistent with reported electrical measurements.},
language = {en},
number = {3},
urldate = {2025-06-22},
journal = {Journal of Applied Physics},
author = {Harima, Hiroshi and Nakashima, Shin-ichi and Uemura, Tomoki},
month = aug,
year = {1995},
pages = {1996--2005},
file = {PDF:/home/chn/Zotero/storage/Y9PU79VE/Harima et al. - 1995 - Raman scattering from anisotropic LO-phonon–plasmon–coupled mode in n -type 4H– and 6H–SiC.pdf:application/pdf},
}
@article{yan_single-defect_2021,
title = {Single-defect phonons imaged by electron microscopy},
volume = {589},
issn = {0028-0836, 1476-4687},
url = {https://www.nature.com/articles/s41586-020-03049-y},
doi = {10.1038/s41586-020-03049-y},
language = {en},
number = {7840},
urldate = {2025-06-22},
journal = {Nature},
author = {Yan, Xingxu and Liu, Chengyan and Gadre, Chaitanya A. and Gu, Lei and Aoki, Toshihiro and Lovejoy, Tracy C. and Dellby, Niklas and Krivanek, Ondrej L. and Schlom, Darrell G. and Wu, Ruqian and Pan, Xiaoqing},
month = jan,
year = {2021},
pages = {65--69},
file = {PDF:/home/chn/Zotero/storage/K3PFH67S/Yan et al. - 2021 - Single-defect phonons imaged by electron microscopy.pdf:application/pdf},
}
@article{_n-sic_2010,
title = {n-{SiC拉曼散射光谱的温度特性}},
volume = {59},
issn = {1000-3290},
url = {https://kns.cnki.net/KCMS/detail/detail.aspx?dbcode=CJFQ&dbname=CJFD2010&filename=WLXB201006094},
abstract = {测量了采用离子注入法得到掺N的n-SiC晶体从100—450K的拉曼光谱.研究了SiC一级拉曼谱、电子拉曼散射谱及二级拉曼谱的温度效应.实验结果表明,大部分SiC一级拉曼峰会随温度升高向低波数方向移动,但声学模红移(峰值位置向低频方向移动)的幅度较光学模小.重掺杂4H-SiC的纵光学声子等离子体激元耦合(LOPC)模频率随温度升高表现出先蓝移(峰值位置向高频方向移动)后红移的变化趋势,表明LOPC模的温度特性不仅会受到非简谐效应的影响,还与实际已离化杂质浓度有关.电子拉曼散射峰线宽随温度升高而增大,强度随温度升高而减弱,但其峰值位置基本不变.二级拉曼谱的红移不如一级拉曼谱明显,但其峰值强度却随着温度的升高显示出明显下降的趋势.},
language = {zh-CN},
number = {6},
urldate = {2025-06-22},
journal = {物理学报},
author = {{韩茹} and {樊晓桠} and {杨银堂}},
year = {2010},
note = {download: 369
album: 基础科学
CLC: O473
dbcode: CJFQ
dbname: CJFD2010
filename: WLXB201006094
CNKICite: 6},
keywords = {温度, 电子拉曼散射, 碳化硅, 纵光学声子等离子体激元耦合模},
pages = {4261--4266},
file = {n-SiC拉曼散射光谱的温度特性:/home/chn/Zotero/storage/M94LRGCT/n-SiC拉曼散射光谱的温度特性.pdf:application/pdf},
}

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#!/usr/bin/env zsh

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= Results and Discussion
我们先讨论无缺陷的情况，再讨论有缺陷的情况。
无缺陷的情况可以解释绝大多数拉曼峰，其它情况从拉曼中的小峰和峰的改变中看出。
// - 无缺陷：
// 我们将声子分为两类，一类是极性比较弱的（18个），一类是比较强的（3个）。
// - 弱极性的声子：
// - TODO: 确认一下最后一次实验中，峰偏移等是否与掺杂有明显关系，以及这个关系与之前是否相同。
// - 强极性的声子：
// - 强极性声子在 Gamma 附近散射谱不连续，它的声子模式由入射光的方向决定。在入射光不沿 z 轴的情况下，使用 C6v 群不再适用。
// - TODO: 写文字
// - 在接近 y 轴入射时，可以看到分裂。这个模式可能对表面敏感。
// - TODO: 佐证它对表面敏感
// - 对于 LO，可能形成 LOPC
// - 有缺陷的情况：
// - TODO: 描述缺陷原子的振动
// - TODO: 计算拉曼张量，描述光谱的可能变化
#include "perfect/default.typ"
#include "defect/default.typ"

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== Phonons with impurities and charge carriers

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== Phonons in Perfect 4H-SiC
总共有 21 个声子模式。我们把它分成 18 个极性很小的和 3 个极性很强的两类。
// There are 21 phonons in total.
// We classified them into two categories: 18 negligible-polar phonons and 3 strong-polar phonons.
The phonons involved in Raman scattering are located in reciprocal space around the #sym.Gamma point,
and the exact positions are determined by the wavevectors of the incident and scattered light.
At each such position, there are 21 phonon modes (degenerate modes are counted as their multiplicity).
We classify these 21 phonons into two categories based on their polarities.
The 18 of 21 phonons are classified into negligible-polar phonons (i.e., phonons with zero or very weak polarity),
for which the effect of polarity can be ignored in the Raman scattering process;
and the other three phonons are strong-polar phonons,
where the polarity gives rise to observable effects in the Raman spectra.
这个分类是有意义的。从微观上讲，它也和原子的振动方向有关。
// This classification make sense.
This classification is based on the fact that
the four Si atoms in the primitive cell of 4H-SiC carry similar positive Born effective charges (BECs),
and the four C atoms carry similar negative BECs (see @table-bec).
In the 18 negligible-polar phonons,
the vibrations of two Si atoms are approximately opposite to those of the other two Si atoms,
and the same holds for the C atoms,
leading to cancellations of macroscopic polarity.
In contrast, in the three strong-polar phonons,
all Si atoms vibrate in the same direction, and all the C atoms vibrate in the opposite direction,
resulting in a strong dipole moment.
#include "table-bec.typ"
#include "non-polar/default.typ"
#include "polar/default.typ"

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=== Phonons with Negligible Polarities
我们用 gamma 点的声子来近似弱极性的声子。
// We investigate phonons at Gamma instead of the exact location near Gamma.
Phonons at the #sym.Gamma point were used
to approximate negligible-polar phonons that participating in Raman processes
regardless of the wavevector of the incident and scattered light.
This approximation is widely adopted (cite) and justified by the fact that,
although the phonons participating in Raman processes are not these strictly located at the #sym.Gamma point,
they are very close to the #sym.Gamma point in reciprocal space
(about 0.01 nm#super[-1] in back-scattering configurations with 532 nm laser light,
which corresponds to only 1% of the smallest reciprocal lattice vector of 4H-SiC,
see orange dotted line in @figure-discont),
and their dispersion at #sym.Gamma point is continuous with vanishing derivatives.
Therefore, negligible-polar phonons involved in Raman processes
have nearly indistinguishable properties from those at the #sym.Gamma point.
#include "figure-discont.typ"
gamma 点处 18 个声子的表示。它们的拉曼张量的形状可以确定，但大小无法确定。
// Representation of these 18 phonons, and the shape of their Raman tensors could be determined in advance.)
Phonons at the #sym.Gamma point satisfy the C#sub[6v] point group symmetry,
and the 18 negligible-polar phonons correspond to 12 irreducible representations of the C#sub[6v] point group:
2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2].
Phonons belonging to the A#sub[1] and B#sub[1] representations vibrate along the z-axis and are non-degenerate,
while those belonging to the E#sub[1] and E#sub[2] representations vibrate in-plane and are doubly degenerate.
Phonons of the B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
In contrast, phonons of the other representations are Raman-active,
and the non-zero components of their Raman tensor
can be determined by further considering their representation in the C#sub[2v] point group (see @table-rep).
These Raman-active phonons are potentially be visible in Raman experiment under appropriate polarization configurations.
However, whether a mode is sufficiently strong to be experimentally visible
depends on the magnitudes of its Raman tensor components,
which cannot be determined solely from symmetry analysis.
#include "table-rep.typ"
// We propose a method to estimate the magnitudes of the Raman tensors of these phonons,
// without first-principle calculations.
// Here we only write out results, details are in appendix.
我们提出了一个新的办法来估计拉曼张量大小。
We propose a method to estimate the magnitudes of the Raman tensors of these phonons based on symmetry analysis.
This approach is founded on the assumption that the change in polarizability induced by atomic displacements in 4H-SiC
is primarily determined by the first- and second-nearest neighbors of the atom and the sign of the atomic charge,
while other factors (mass, bond length, etc.) only have small contributions.
Consequently, the Raman tensors of the calculated phonon modes can be estimated
before additional first-principles computations (see appendix for details),
and the results are summarized in @table-nopol.
The parameters $a_i$ exhibit significantly larger absolute values compared to $epsilon_i$, $eta_i$, and $zeta_i$,
indicating the E#sub[2] mode at 756.25 cm#super[-1] in simulation (mode 8)
possess a much higher Raman intensity than the others.
我们使用第一性原理计算得到了拉曼张量的大小，并与我们的结果进行了比较。
The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods,
and the results are compared with both experimental data and theoretical predictions (@table-nopol).
The calculated phonon frequencies show good agreement with experimental data with a slight underestimation of 2-5%,
where the error may be attributed to the underestimation of interatomic forces by the PBE functional (cite).
The calculated Raman tensors are also consistent with experimental and theoretical results.
Among negligible-polar modes, the E#sub[2] mode at 776 cm#super[-1] in experiment (mode 8)
exhibits the highest Raman intensity,
followed by four modes with lower intensities that are also experimentally visible,
including the E#sub[2] modes at 195.5 cm#super[-1] (mode 1) and 203.3 cm#super[-1] (mode 2),
the E#sub[1] mode at 269.7 cm#super[-1] (mode 3), and the A#sub[1] mode at 609.5 cm#super[-1] (mode 6).
The E#sub[1] mode calculated at 746.91 cm#super[-1] (mode 7)
and the E#sub[2] mode calculated at 756.25 cm#super[-1] (mode 9)
are predicted to have much weaker Raman intensities and are located close to the most intense mode (mode 8),
making them indistinguishable in experimental spectra.
Additionally, the A#sub[1] mode calculated at 812.87 cm#super[-1] (mode 10)
exhibits a very weak Raman intensity in the basal-plane polarized configurations (xx and yy, only 0.01)
but shows an observable intensity when the polarization is along the z-axis (1.78).
Since most Raman experiments are performed in a back-scattering configuration with light incident along the z-direction
(i.e., with in-plane polarization)
and with photon energies much lower than the band gap,
this mode is typically not observed (cite).
However, it should become detectable if the incident light has a polarization component along the z-direction
(as in our experiment),
or when the excitation wavelength approaches resonance conditions (cite).
其它峰在其它章节中解释。
Besides, there are other peeks in the experiment.
The peek at 796 and 980 are caused by strong-polar phonons which will be discussed later.
Besides, there are small peeks at xxx,
which could not be explained in perfect 4H-SiC and will be discussed in the next section.
// TODO: 将一部分 phonons 改为 phonon modes
// 在论文中我们这样来称呼：phonon 对应某一个特征向量，而 modes 对应于一个子空间。
// 也就是说，简并的里面有两个或者无数个 phonon，但只有一个 mode
#include "table-nopol.typ"
#include "figure-raman.typ"
// TODO: 解释为什么 E1 可以看到

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#figure(
image("/画图/声子不连续/embed.svg"),
caption: [
(a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path.
Gray lines represent negligible-polar phonon modes,
while colored lines indicate strong-polar phonon modes.
The green, red and blue lines indicate the mode along the z-direction, y-direction and x-direction, respectively.
Along A-#sym.Gamma path, strong-polar modes along x- and y-directions are degenerated,
showing as a single purple line.
(b) Magnified view of the boxed region in (a).
The orange dashed lines mark the phonon wavevectors involved in Raman scattering
with incident light along the z- and y-directions.
],
placement: none,
)<figure-discont>

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#figure(
image("/画图/拉曼整体图/main.svg"),
caption: [
(a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path.
Gray lines represent negligible-polar phonon modes,
while colored lines indicate strong-polar phonon modes.
(b) Magnified view of the boxed region in (a).
The orange dashed lines mark the phonon wavevectors involved in Raman scattering
with incident light along the z- and y-directions.
]
)<raman>

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#page(flipped: true)[#figure({
set text(size: 9pt);
set par(justify: false);
let m(n, content) = table.cell(colspan: n, content);
let m2(content) = table.cell(colspan: 2, content);
let m3(content) = table.cell(colspan: 3, content);
let A1 = [A#sub[1]];
let A2 = [A#sub[2]];
let B1 = [B#sub[1]];
let B2 = [B#sub[2]];
let E1 = [E#sub[1]];
let E2 = [E#sub[2]];
table(columns: 22, align: center + horizon, inset: (x: 3pt, y: 5pt),
m2[*Number of Mode*],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[1], m2[2], m2[3], [4], [5], m2[6], m2[7], m2[8], m2[9], m2[10], [11], [12],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[*Vibration Direction*], [x], [y], [x], [y], [x], [y], m(4)[z], [x], [y], [x], [y], [x], [y], m(4)[z],
table.cell(rowspan: 2)[*Representation*],
[C#sub[6v]], m2(E2), m2(E2), m2(E1), B1, B1, m2(A1), m2(E1), m2(E2), m2(E2), m2(A1), B1, B1,
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
[C#sub[2v]], A2, A1, A2, A1, B2, B1, B1, B1, m2(A1), B2, B1, A2, A1, A2, A1, m2(A1), B1, B1,
// TODO: 重新检查数据是否正确（主要是正负号）
table.cell(rowspan: 4)[*Raman Tensor*],
[Non-zero components],
// E2 E2 E1 2B1 A1
[xy], [xx, -yy], [xy], [xx, -yy], [xz], [yz], m2[None], [xx, yy], [zz],
// E1 E2 E2 A1 2B1
[xz], [yz], [xy], [xx, -yy], [xy], [xx, -yy], [xx, yy], [zz], m2[None],
[Analysis result (a.u.)],
// E2 E2 E1 2B1
m2[$2epsilon_2-2zeta_2-4eta_2$], m2[$2epsilon_2-2zeta_2$], m2[$-2epsilon_1-2zeta_1$], m2[-],
// A1 E1 E2
[$-2epsilon_5$], [$-2epsilon_6$], m2[$-2epsilon_1+2zeta_1$], m2[$8a_2+2epsilon_2-2zeta_2-4eta_2$],
// E2 A1 2B1
m2[$-2epsilon_2+2zeta_2$], [$-2zeta_5$], [$-2zeta_6$], m2[-],
[Calculation result (a.u.)],
// TODO: 改正正负号
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[0.17], m2[1.13], m2[2.43], m2[-], [2.83], [1.79], m2[0.09], m2[88.54], m2[0.50], [0.01], [1.78], m2[-],
[Experiment result (a.u.)],
// TODO: 填充
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[], m2[], m2[], m2[-], [], [], m2[Invisible], m2[], m3[Invisible], [], m2[-],
table.cell(rowspan: 3)[*Frequency*],
[Simulation (cm#super[-1])],
// E2 E2 E1 2B1 A1
m2[190.51], m2[197.84], m2[257.35], [389.96], [397.49], m2[591.90],
// E1 E2 E2 A1 2B1
m2[746.91], m2[756.25], m2[764.33], m2[812.87], [885.68], [894.13],
[Experiment (cm#super[-1])],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[195.5], m2[203.3], m2[269.7], m2[-], m2[609.5], m2[Invisible], m2[776], m(3)[Invisible], [839], m2[-],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
[Error (%)], m2[2.6], m2[2.7], m2[4.6], m2[-], m2[2.9], m2[-], m2[2.5], m(3)[-], [3.1], m2[-],
table.cell(rowspan: 2)[*FWHM* (cm#super[-1])],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
[Simulation], m2[0.08], m2[0.09], m2[0.08], m2[-], m2[0.61], m2[3.97], m2[4.62], m2[4.01], m2[0.89], m2[-],
// TODO: 选取合适的实验并填充数据
[Experiment, zxxz],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[2.61], m2[2.09], m2[1.98], m2[-], m2[2.64], m2[Invisible], m2[3.27], m3[Invisible], [], m2[-],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[*Polarity*], m(4)[None], m2[Weak], m2[None], m(4)[Weak], m(4)[None], m2[Weak], m2[None],
)},
caption: [Negaligible-polarized Phonons at $Gamma$ Point.],
)<table-nopol>]

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#figure({
set text(size: 9pt);
set par(justify: false);
let m2(content) = table.cell(colspan: 2, content);
let A1 = [A#sub[1]];
let A2 = [A#sub[2]];
let B1 = [B#sub[1]];
let B2 = [B#sub[2]];
let E1 = [E#sub[1]];
let E2 = [E#sub[2]];
table(columns: 7, align: center + horizon,
[*Representations in C#sub[6v]*], A1, B1, m2(E1), m2(E2),
[*Representations in C#sub[2v]*], A1, B1, B2, B1, A2, A1,
[*Vibration Direction*], [z], [z], [x], [y], [x], [y],
[*Raman Tensor*],
[$mat(a,,;,a,;,,b)$], [$0$], [$mat(,,a;,,;a,,;)$], [$mat(,,;,,a;,a,;)$], [$mat(,a,;a,,;,,;)$], [$mat(a,,;,-a,;,,;)$],
[*Raman scatter Intensity* #linebreak() (polarization of incident and scattered light)],
[xx/yy: $a^2$ #linebreak() zz: $b^2$ #linebreak() others: 0], [0],
m2[xz/yz: $a^2$ #linebreak() others: 0], m2[xx/xy/yy: $a^2$ #linebreak() others: 0],
)},
caption: [
Irreducible representations and raman tensors of phonons in 4H-SiC.
],
placement: none,
)<table-rep>

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=== Strong-polar Phonons
沿着不同方向入射的话，强声子的模式是不同的。
// 在半导体的极性声子模式中，原子间存在长距离的库伦相互作用，导致散射谱在 Gamma 附近不再连续（引用），如图中的彩色线所示。
// 这导致不同方向的入射/散射光的声子模式不同。
// 具体来说，当入射光/散射光沿着 z 方向时，起作用的是 A-Gamma 线上的声子模式（图中的左半边的橘线），它们适用于群 C6v。
// 这时会有一个 E1 模式（TO，振动方向在面内）和一个 A1 模式（LO，沿 z 振动）。
// 而当沿着 y 方向入射时，起作用的是 Gamma-K 线上的声子模式（图中的右半边的橘线），它们不再适用于群 C6v，而只适用于群 C2v；
// 它会分裂成沿x、y、z 方向的三个声子模式（图中的右半边的蓝线），它们分别对应于群 C2v 的 A1、B1 和 B2 表示 TODO: 确认这个几个表示的名字。
// 若考虑到到入射光不是严格沿着 z 方向，而是有一个小的角度（例如 10 度），则此时有一个声子模式沿着 x 方向，另外两个声子模式则为 y-z 两个方向的混合。
// (没有在图上表示)
Strong-polar phonon modes caused by different incident light directions are different,
due to long-range Coulomb interactions between atoms in semiconductors,
showing discontinuity in the scattering spectra near the #sym.Gamma point (see @figure-discont).
For incident light propagating along the z direction (phonon modes on the A-#sym.Gamma line),
symmetry of C#sub[6v] point group applies and leading to two modes (two peeks in Raman spectra),
including an E#sub[1] mode (pink line in @figure-discont, vibration in-plane)
and an A#sub[1] mode (green line in @figure-discont, vibration along z-direction).
When the light is incident along other directions, symmetry in plane was broken and C#sub[6v] symmetry no longer holds,
and there will be three phonon modes in theory.
For example, when the light is incident along the y direction (phonon modes on the #{sym.Gamma}-K line),
symmetry of C#sub[2v] applies and three modes exist in dispersion curves,
including an A#sub[1] mode (green line in @figure-discont, vibration along z direction),
a B#sub[2] mode (blue line in @figure-discont, vibration along x direction),
and a B#sub[1] mode (red line in @figure-discont, vibration in y direction).
When the light is incident along a direction between z and y,
three phonon modes will exist, but vibration in the mixed direction.
实验发现的确是这样的。
Many Raman experiments on 4H-SiC with incident light along the z direction have observed two peaks.
However, no experiments have reported three peaks with incident light along other directions.
In our experiment, we found the third, and it satisfied properties we expected.
In our experiments, we found that the third peak only appears when focusing inside the sample.
// 我们预测，随着入射方向偏移，LO 峰会向着高频方向移动。此外，我们也注意到 LO 也会与载流子产生影响。
// 在 n 型半导体中，LOPC 模式将代替 LO 模式；在 p 型半导体中，LO 模式仍然单独存在，但它的半高宽会受到载流子浓度的影响。
#include "table-pol.typ"

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#page(flipped: true)[
#figure({
set text(size: 9pt);
set par(justify: false);
let m(n, content) = table.cell(colspan: n, content);
let m2(content) = table.cell(colspan: 2, content);
let m3(content) = table.cell(colspan: 3, content);
let A1 = [A#sub[1]];
let A2 = [A#sub[2]];
let B1 = [B#sub[1]];
let B2 = [B#sub[2]];
let E1 = [E#sub[1]];
let E2 = [E#sub[2]];
let NA = [Not Applicable]
let lopc = [Yes#linebreak() (LOPC)];
let overf = [Yes#linebreak() (overfocused)];
table(columns: 19, align: center + horizon, inset: (x: 3pt, y: 5pt),
m2[*Incident Direction*], m(4)[z], m(4)[y], m(9)[between z and y, 20#sym.degree to z],
m2[*Vibration Direction*],
// TODO: check LO-TO mixed
[TO (x)], [TO (y)], m2[LO (z)], m3[TO (z)], [TO (x)], [LO (y)], m3[TO (y-z mixed)], [TO (x)], m(4)[LO (y-z mixed)],
table.cell(rowspan: 2)[*Representation*],
[C#sub[6v]], m2(E1), m2(A1), m(13, NA), [C#sub[2v]], B2, B1, m2(A1), m2(A1), B2, B1, m(9, NA),
table.cell(rowspan: 2)[*Raman Tensor*],
[Non-zero components],
[xz], [yz], [xx, yy], [zz], // z
[xx, yy], [zz], [xz], [yz], // y
[xx], [yy], [yz], [zz], [xz], [xx], [yy], [yz], [zz], // 25 y&z
[Simulation Result (a.u.)],
// TODO: raman intensity, or raman tensor?
m2[53.52], [58.26], [464.69], // z
[58.26], [454.09], [53.52], [53.55], // y
m2[53.71], [3.20], [425.98], [53.56], m2[3.60], [50.36], [27.99], // 45 y&z
m2[*Visible in Common Raman Experiment*], m(17)[Yes],
m2[*Wavenumber (Simulation) (cm#super[-1])*],
// z y 45 y&z
m2[776.57], m2[933.80], m2[761.80], [776.57], [941.33], m(4)[762.76], [776.57], m(4)[940.86],
m2[*Electrical Polarity*], m(17)[Strong]
)},
caption: [Strong-polarized phonons near $Gamma$ point],
)
]

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#figure({
set text(size: 9pt);
set par(justify: false);
table(columns: 4, align: center + horizon,
table.cell(colspan: 2, rowspan: 2)[], table.cell(colspan: 2)[BEC (unit: |e|)], [x / y direction], [z direction],
table.cell(rowspan: 2)[Si atom], [A/C layer], [2.667], [2.626], [B layer], [2.674], [2.903],
table.cell(rowspan: 2)[C atom], [A/C layer], [-2.693], [-2.730], [B layer], [-2.648], [-2.800],
)},
caption: [
Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC, calculated using first principle method.
],
placement: none,
)<table-bec>

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# 关键词
“SiC 中的**声子**”作为文章的中心，而不是拉曼、SiC 的表征 etc。因为这样可以将所有东西串起来。
# Introduction
分为三段：
```mermaid
graph
A[第一段前半：<b>SiC</b> 很好的性质、重要的应用场景]
--> B[第一段后半：SiC 中的<b>声子</b>很重要，它对材料性质有怎样的影响，或者可以反映出怎样的材料性质（作表征）]
--> C[第二段：关于 SiC 中的声子，有哪些<b>已有</b>的研究，以及这些研究的<b>不足</b>]
--> D[第三段：本文做了什么事情，尤其强调第一次做了什么]
```
# Method
# Results and Discussion
我们首先考虑没有缺陷的 4H-SiC 中的声子（不考虑缺陷/掺杂的直接影响，但会考虑费米能级/载流子的影响），然后再考虑缺陷的影响。
## 无缺陷的情况
拉曼散射对应的声子是处在 Gamma 点附近（但不严格在 Gamma 点）的声子。根据这些声子在拉曼散射实验中的可见性，我们将它们分为两类讨论：
1. 在拉曼实验中不可见的声子（既包括没有拉曼活性的声子，也包括有拉曼活性但散射强度太弱的声子）。
2. 在拉曼实验中可见的声子。
4H-SiC 在 Gamma 点的声子共有 21 个模式，这些模式对应于点群 $\mathrm{C_{6v}}$ 的 14 个表示（$\mathrm{3A_1+4B_1+3E_1+4E_2}$，其中 $\mathrm{A_1}$、$\mathrm{B_1}$ 为一维表示，对应于无简并的声子；$\mathrm{E_1}$、$\mathrm{E_2}$ 为二维表示，对应于二重简并的声子）。在拉曼实验中，起作用的声子并不严格在 Gamma 点；但大多数声子的色散谱在 Gamma 点连续且导数（斜率）为零，因此大多情况下可以沿用这个分类，少数情况我们稍后会专门讨论。
==这两段话的顺序，是现在这样安排比较好，两段颠倒一下比较好？这样安排可以让理解的难度循序渐进，并且是把重要的事情（分类）写在前面，把不重要的（具体的性质）放在后面；如果颠倒的话，突然把群表示论糊到读者头上，读者会不会蒙？==
### 拉曼实验中不可见的声子
对应于 $\mathrm{B_1}$ 表示的四类声子是不具有拉曼活性的，它们不会对入射光造成拉曼散射，实验中我们也的确没有在这里看到散射峰。此外，它们也不具有红外活性，因此也应该（should）无法通过红外吸收谱研究。
| 表示 | 频率（THz） | 波数（$\mathrm{cm^{-1}}$）计算值 | 波数（$\mathrm{cm^{-1}}$）实验值 | 拉曼强度 | 实验中是否可见 |
| :------------: | :---------: | :------------------------------: | :------------------------------: | :------: | :------------: |
| $\mathrm{B_1}$ | 11.70 | 389.96 | - | 0 | 否 |
| $\mathrm{B_1}$ | 11.92 | 397.49 | - | 0 | 否 |
| $\mathrm{B_1}$ | 26.57 | 885.68 | - | 0 | 否 |
| $\mathrm{B_1}$ | 26.82 | 894.13 | - | 0 | 否 |
==“拉曼强度（拉曼张量的大小）”“散射强度（预测的散射峰的面积）”这两个概念/数值有微小的差别，我需要确认一下它们对应的英文是什么。这里指的是前者，表格中填写的是对应分量的平方或平方和。前者到后者需要乘以一个系数，不同峰的系数不同，但在本文的实验条件下，差别只有大约 10%，在本文中没必要详细讨论，保证概念的使用正确即可。==
此外，根据第一性原理计算，还有 7 个拉曼活性的模式散射强度比较小。这些模式对应于 4 个表示，它们在拉曼散射谱上本应该表现为 4 个较小的散射峰。然而，其中两个恰好位于 4H-SiC 的主散射峰（最高的散射峰）附近，主峰的展宽导致它们在通常的拉曼光谱上无法分辨。这两个峰可能可以在低温拉曼散射实验看到。除此此外，这个 $\mathrm{E_1}$ 也具有较弱的极性，可能可以在红外吸收谱中看到。
我们将这 4 个表示的信息列于下表。为了对比，同时也列出主散射峰的信息。
| 表示 | 频率（THz） | 波数（$\mathrm{cm^{-1}}$）计算值 | 波数（$\mathrm{cm^{-1}}$）实验值 | 拉曼强度（xx/xy/xz/zz） | 实验中是否可见 |
| :--------------------: | :---------: | :------------------------------: | :------------------------------: | :---------------------: | :------------: |
| $\mathrm{E_2}$ | 5.72 | 190.51 | 195.5 | 0.17/0.17/0/0 | 是 |
| $\mathrm{E_1}$ | 22.41 | 746.91 | - | 0/0/0.09/0 | 否 |
| $\mathrm{E_2}$（主峰） | 22.69 | 756.25 | 776 | 88.70/88.54/0/0 | 是 |
| $\mathrm{E_2}$ | 22.93 | 764.33 | - | 0.50/0.51/0/0 | 否 |
| $\mathrm{A_1}$ | 24.39 | 812.87 | 839 | 0.01/0/0/1.78 | 是 |
==“主散射峰”这个名词是否不太合适？通用的应该怎么说？==
==这几个峰比较小的原因应该是可以从群表示论中找到的。应该只需要将拉曼张量的贡献具体到每个原子，就可以得到结果。==
综上所述，在通常的拉曼实验中不可见的声子共有 $\mathrm{4B_1+E_1+E_2}$。
### 拉曼实验中可见的声子
Si-C 键为极性键，Si 可以视为正电荷中心、C 为负电荷中心；原子核振动时，电子也会随之振动，这导致了一些声子模式带有电极性。
具体来说，在 4H-SiC 的原胞中，ABCS 四层中的四个 Si 原子所带电荷差别不大，四个 C 原子所带电荷同样相近（见下表）。因此，我们将拉曼可见的声子分为两类：
| 原子 | 层 | Born 有效电荷（xx/xy/xz/zz，单位为元电荷） |
| :--: | :--: | :----------------------------------------: |
| Si | A/C | 2.66680/0/0/2.62625 |
| Si | B | 2.67424/0/0/2.90347 |
| C | A/C | -2.69310/0/0/-2.72956 |
| C | B | -2.64794/0/0/-2.80017 |
* 没有极性，或者极性较弱的模式。这包括 $2\mathrm{A_1} + \mathrm{E_1} + 3\mathrm{E_2}$，共 6 个表示（6 个峰）、10 个模式。对于这些模式，原胞中的四个 Si 中的两个的振动方向总是与另外两个相反，C 也是如此；这使得原子振动造成的极性互相抵消，整个模式没有极性（$3\mathrm{E_2}$）或者极性较弱（$2\mathrm{A_1}+\mathrm{E_1}$）。
* 极性较强的模式。在这些模式中，所有的 Si 原子都同向振动，所有的 C 原子都沿相反的方向振动；原子振动造成的极性互相叠加，整个模式具有较强的极性。这包括了三个模式，点群 $\mathrm{C_{6v}}$ 不再适用于分析这三个模式。
我们将这些模式与实验上的拉曼峰关联起来，如图所示。接下来我们分别详细讨论这两类。
![](../画图/拉曼整体图/main.svg)
#### 无极性或弱极性声子
这些声子的特征向量（各个原子的振幅、方向）、频率与入射光的方向基本无关。我们将它们的信息列于下表。
| 表示 | 频率（THz） | 波数（$\mathrm{cm^{-1}}$）计算值 | 波数（$\mathrm{cm^{-1}}$）实验值 | 拉曼强度（xx/xy/xz/zz） |
| :------------: | :---------: | :------------------------------: | :------------------------------: | :---------------------: |
| $\mathrm{E_2}$ | 5.72 | 190.51 | 195.5 | 0.17/0.17/0/0 |
| $\mathrm{E_2}$ | 5.94 | 197.84 | 203.3 | 1.13/1.13/0/0 |
| $\mathrm{E_1}$ | 7.72 | 257.35 | 265.7 | 0/0/2.43/2.43 |
| $\mathrm{A_1}$ | 17.76 | 591.90 | 609.5 | 2.83/0/0/1.79 |
| $\mathrm{E_2}$ | 22.69 | 756.25 | 776.3 | 88.70/88.54/0/0 |
| $\mathrm{A_1}$ | 24.39 | 812.87 | 839 | 0.01/0/0/1.78 |
我们可以看到，计算与实验的误差大约为2-3%。因此，在之后的内容中，我们将不比较
* 计算了**无缺陷/掺杂 4H-SiC 声子**信息，并与实验对照。本小节，我们先不管缺陷、掺杂、载流子等等，只考虑完美的 4H-SiC 中原子如何振动、对应的拉曼光谱长什么样子。
* 这一部分相当于做了这三件事：
* 整合梳理已有文献中的信息；
* 第一性原理计算并与实验对照，讨论它们在哪些方面能对得上、哪些方面对不上，明确每个峰是哪个模式贡献的，并将其振动模式可视化；
* 详细讨论了极性声子模式的频率受到入射/散射光方向的影响。
* 4H-SiC 总共有 **21 个声子模式**（去掉平移，$\mathrm{3A_1+4B_1+3E_1+4E_2}$），我们将它分为**四类**：
* **没有拉曼活性的模式**：在拉曼光谱上看不到。这包括 $\mathrm{C_{6v}}$ 群的 $4\mathrm{B_1}$。接下来解释“$\mathrm{C_{6v}}$ 群的 $\mathrm{B_1}$”是什么意思（这些不写到论文里，只是在组会上解释一下）。
* 4H-SiC 中，$\Gamma$ 点和 $\mathrm{\Gamma-A}$ 路径上的声子（即沿 z 方向入射和散射）对应群 $\mathrm{C_{6v}}$。这个群的 $\mathrm{B_1}$ 表示没有拉曼活性。
* 但其它群的 $\mathrm{B_1}$ 表示可能有拉曼活性（稍后会看到一个例子），不同群的同名表示不是同一个东西，只是重名了而已，它们之间没有关系。
* 其它方向入射的拉曼光谱对应的声子严格来说，没有这么高的对称性；但大多时候，沿用 $\mathrm{C_{6v}}$ 群仍然是合适的，少数情况会特别说明。
* 例如，$\mathrm{\Gamma-M}$ 路径上的声子（即沿 y 方向入射和散射）严格来说只能对应于群 $\mathrm{C_{2v}}$。但大多数模式在 $\mathrm{\Gamma}$ 点附近是连续的，不同方向的拉曼散射对应的声子是非常接近 $\mathrm{\Gamma}$ 点、但分布在 $\mathrm{\Gamma}$ 点的不同方向上的声子。因为连续，所以沿不同方向离开 $\Gamma$ 点很短的距离，导致的频率变化是可以忽略不计的。少数几个极性强的声子可以造成长程的库伦相互作用，导致频率在 $\mathrm{\Gamma}$ 点附近不连续，导致拉曼入射方向会影响声子的频率，这种情况下我会特别说明。
* 虽然**有拉曼活性**，但通过第一性原理计算得知它的**散射强度太弱**，在通常的拉曼实验中看不到（被附近的散射峰覆盖）：这包括 $\mathrm{E_1} + \mathrm{E_2}$，共 4 个模式。
* 按照计算，这两个峰的散射强度只有附近那个峰的几千还是几万分之一，所以看不到（==具体数值下周给==）。
* **拉曼活性且散射足够强**，并且**极性不强或者没有极性**，即频率（峰位）不随入射/散射光的方向改变而改变。这包括 $2\mathrm{A_1} + \mathrm{E_1} + 3\mathrm{E_2}$，六个峰、十个模式，它的位置在图中标出来了。
* 计算的峰位（波数）总是比实验低一些（不超过 5%）。例如，最高的峰实验值为776，但计算值是 756。因此之后研究 797 附近的峰的位置时，总是拿 776 这个峰对齐后再研究（即在计算值上 + 20）。
* **拉曼活性且散射足够强**，并且**极性较强**，峰的位置与拉曼入射方向有关。对于 z 方向入射/散射，它是 $\mathrm{C_{6v}}$ 群的 $\mathrm{A_1}+\mathrm{E_1}$，两个峰、三个模式；当入射/散射方向偏离 $z$ 轴时，$\mathrm{E_1}$ 会分裂成两个峰。例如，对于 y 方向入射/散射，$\mathrm{E_1}$ 会分裂为 $\mathrm{C_{2v}}$ 群的 $\mathrm{A_1} + \mathrm{B_1}$，它们都是拉曼活性的，不加偏振片时，两个峰都能看到，加上偏振片能把两个中的任意一个滤出来。==示意图下周给==。==这里缺一个实验（加偏振片露出来，不加偏振片就两个都能看到，的对比）==。可以看紫色的这条线。
* **这里有一个与计算明显不符合的地方**：计算中，它们会分裂大约 15 个波数；但实验上只分裂了 3 个波数。我的猜想是，有可能有下面这些原因：
1. 实验中不是完全沿着 y 轴入射。但我计算发现，即使是偏离 20 度，甚至 45 度斜入射（沿着 z 和 y 中间入射），分裂都会大于 10 个波数；因此应该不是这个原因。
2. ==载流子的影响==（这实际上是下一节的内容）。我实验用的是衬底（大概 $10^{19}$ n 掺杂），自由载流子很大程度上屏蔽了长程库伦相互作用，导致分裂明显变小。要证实这个猜想，要：
1. 计算带电荷的声子。正在算，下周组会能出一部分或者全部结果。（不同浓度、不同类型的载流子，在不同入射方向下，会分裂多少）。
2. 实验测外延层的信号。
3.
* 我计算了这些模式在各种偏振下的散射强度，与实验结果符合得比较好。（对比给读者看）
* ==怎么去表达“符合得比较好”这个结果==？这里有两个问题：
* 六个峰、五个偏振方向，实验、计算各 30 个数值，数值比较多。
* 实验时无法做到绝对的沿着某个方向偏振（总是会有一点倾斜），导致某些理论上在某个偏振下看不到的峰，在实验中实际上能看到。
* 我的计划是把倾斜的角度也加入计算，但可能要尝试几次才能得到比较好的结果（因为不知道具体是倾斜10度还是20度还是5度）。
* 声子与载流子的相互作用，使用 LOPC 估计 n 和 p 掺杂的浓度。（？）
* 计算了缺陷和掺杂的振动模式，解释了光谱上的一些小峰的来源（来自于掺杂）。